Joseph Leo Doob, 1962-2003

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Title:
Joseph Leo Doob, 1962-2003
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English
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Doob, Joseph Leo
Chung, Kai Lai
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Box: 1
Folder: Joseph Leo Doob, 1962-2003

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Mathematics -- History -- 20th century

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University of Florida
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Congratulations to J. L. Doob


We are very happy to congratulate you on winning

the National Medal of Science. This award is an appropriate
recognition of the profound and lasting influence you have

Been exercising on probability and mathematical statistics for more
than forty years. To the long-standing appreciation of your

- colleagues the world over, there has now been added this high

honor.

Heartiest congratulations and best wishesI













Professor Kai Lai CM
Department of Statistics
Stanford University
Stanford
California




April 27, 1962
Dear Chung: Now J understand something that has been bothea
ing me. I object to your definition of.Borel meas. of a pr
Tt would be more natural to define it using not all meas.
cb-sets but only the field generated by the x(t)'s. In any
theorem that there exists a version, this is what is proved
and the use of all meas. sets may cause accidents. I think
that this will make life easier. Incidentally you should
state things like: if a< 6 are stopping times then 6-a is
a stopping time for the post a process. This is in line mri
my letter on how you should generalize your work.


xp-;P






HEART DISEASE


THIS SIDE OF CARD IS FOR


Professor Kai Lai Chung
Department of Statistics
Stanford University
Stanford
California





March 30, 1962
Dear Chung:
Let F be the class of Borel subsets of [0,1]. Let
G be the class of subsets of the interval whose indicator
n -n
functions have period 2 and let'G decrease to G.
Then FvG is the class of all subsets of the interval. Can
you shownthat FvG is smaller? I can't yet, but it looks
plausible.





ff ^'" / '











10/23/1963
Dear Doob:
I have not read your writing in the MS but spent some
time to correct the errors you found. It was harder to
figure out from what you said what was wrong than to correct
it. I still don't see what is garbled at one place except
perhaps a verb should be in the singular case. The place rc
*i4 my assertion of some collection.being a Borel field is
incorrect seems correct if the Borel field is on (ta c). I will
mark these places in the mimeo. version later.

You were right that I cannot prove strict optionality of
a+h as asserted, but I cannot decide if this is true when both
of them are positive and strictly optional. Can you decide this
one? And what about the other propositions I wrote you about?

~Mejer descended on us with his whole family. He showed me
his book MS which contained some of the stuff we have, but only
the "well-knovwn" ones. He said he thought of OAlast result
about-a right continuous family of optional times but did not
know how to prove it. That surprised me as well as your simpli-
fication which I have yet to read. In any case, you should ask
him what he wishes in the matter of acknowledgment of simulta-
neity.
I believe so long as mathematics department and statistics
department coexist double standard cannot be avoided. Indeed.
I told Day the other day that the creation of so many dept of
stat. is purely a matter of human and political consideration, not
a scientific one. Otherwise the only reasonable way is to have
the better mathematicians (I will draw the line say short of
Hodges) in the math dept and let the stat spread among all the
schools on campus which needA it. The trouble is of course some
big stat frog must have its own stinking pond to puddle in.
In the case of my PH. D. student* had he done a lot more exposi-
tory work (some~4OA new proofs* and many coinections,...) I would
have accepted it without asking anyone's opinion, but the case
is obvious since he has not even done that, apart from quite a
few baderrors. Thanks for the letter anyway which I quoted from
anonymously.











August 5, 1964
Dear Doob:

I sent the corrections to Hunt. It is just like him to worry
about possible misapprehension of what is correct (Prop.,.l) and not
so mauhabout impossible apprehension of what is not incorrect (meaning
his own style).

I have been proving some beautiful theorems about boundaries. They
are becoming very transparent indeed. What was foolish about my Acta
Math. paper was I had been barking up the same wrong tree as Feller did
in using that dual boundary. Since it really did not make sense except
in particular cases of course it did not get- far. My new approach omits
any dual boundary and xPi visualizes the boundary points as "banners"
under which the states may stand, and make the important distinction be-
tween recurrent and nonrecurrent banners and basically reduce everything
to "trapping" (=absorbing) banners. Now the boundary points behave so
much like states and all the analytical mysteries disappear. Here is one
interesting thing. As you know the so-called Blumenthal 0-1 law does not
apply to the strong Markov property as proved in my paper for exits to
boundary, but the follwoing case is valid Call a boundary point "sticking"
iff upon reaching it there is positive probability that you will be stuck
with it a while. Then the prob. that a given boundary point be sticking
is either 0 or 1. I don't know how to prove this using the Borel-Cantelli
type of argument since there is no well-ordering of the sticking part, but
it falls out easily from an analytic formula linkingwith the local time
infinitely divisible law. If you can give a direct proof of this using
your great definitions of Martin boundary, I will send you a nickel.
I won't rewrite my book but now I wonder if I should add another Ap-
pendix giving the boundary stuff. It can contain the fisrt fiuw sections
of my Acta paper and another twenty five pages new material to finish
the case of finitely many boundary points. I shall swear off the subject
then.-This means an addition of some 50 pages to the book. What do you
think? I have not decided to do it yet and may content with the correc-
tions I sent yous but add at least 50 percent to it.


Sincerely yours,







UNIVERSITY OF ILLINOIS
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801

February 1, 1968
Dear Chung:
Meyer evidently doesn't know how the other half lives. As I
remember the Levy household it was the kind of place where when a
telegram arrives it is carried in to the master by a maid, and
probably on a silver tray. So the master may be disturbed by constant
bell ringing but certainly does not have to get up!
Don't rub it in that I am getting old. Who would know it better
than I? When I see the amount of material Dolgans knows and compare
it with what I manage to impart to my students it makes me want to
retire right now. Talking about getting old: Halmos has accepted the
job of head at Honolulu! And while on the subject of moving: it
appears that Robbins is going back to Columbia and that Chow will
probably go with him. The big news at the San Francisco meeting was
that the expansion may be over. There are far fewer jobs available
now for young men.
I don't agree with you about the compactifications. I feel that
they offer new insights, also new problems. It seems to me that the
natural way to do the things that you and Feller and more recently
Dynkin have been doing assuming finitely many boundary points is to
assume corresponding things about the proper compactification. (This
is not a trivial change, partly because one must decide whether it makes
much difference to assume only finitely many new boundary points or
finitely many new limit points, which may be old points. Incidentally
this shows that Knapp was right in KSK to stress my type of Martin
boundary in which points in the space may be on the boundary.) I
am going to put a student on these things. Last night I tried for a
while to see just what my old return from infinity means and even that
was not easy to pin down, that is to describe exactly what the boundary
looks like.
Some of what I wrote last time was nonsense, as I found out when,
I started to write things up carefully. ._ m

tm at my decomposition of K was vd nonsense since
each set B is a singleton.
a
Here is an example that is illuminating about boundaries. Suppose
e,&- _everything is stable and nice and one goes back from infinity using a
< fixed distribution as in my old work. Then using the Martin compactification
there is right continuity and left limits. The Martin compactification
may have introduced lots of new points. But in terms of the compactification
I and Meyer use it can be shown that the left limit at a blowup is only
a single point. In other words the fine distinctions of the Martin
boundary are lost. This illustrates my point that the requirements of
any compactification should be axiomatized, with an existence theorem
of course for any set of requirements. Some results are then independent
of the compactification: e.g. only one integer is taken on arbitrarily
close to a parameter value, on one side; some depend on the compactification,
e.g. right continuity of paths, but the latter may be true for quite
different compactifications which are useful for different purposes.
That Tokyo symposium is a year from this spring! I missed the date
and it never occurred to me that plans were made so far ahead.
I am not sure whether it means that you are younger or older than i
if you have turned to French novels whereas I have turned to recorder
playing. I suppose it depends on the French novels, in particular on
whether they are illustrated. Don'tlet anyone push you onto Proust. A








colleague in the French department suckered me into reading his
Remembrance etc., bringing me a couple of volumes or so of his
French edition at a time and giving me double talk when I asked how
many more were to come. By the time I had finished the 16 or so volumes
I was thoroughly bored. It doesn't take that long to analyze time
in a math paper, and parts of the novel were about that interesting.
It is a deal: jet fare plus 500 here for something under 2 weeks.
Let me know well ahead of time when you are coming and I'll get you
a room at the Union, or at the Urbana-Lincoln if you prefer (It is
part of a new development and you can sit sort of outside and watch
the crowds go past in a large rather attractive covered arcade.)
Political opinion is really changing a bit now. There may be some
hope that Johnson will be forced to listen, but I am afraid that all
the talk about stopping the bombing is off the point. Ho is no more
interested in stopping short of guaranteed victory than Johnson is.
If Johnson actually stops the bombing and nothing comes of it the
doves will be discredited in the public eye. Stopping the bombing
will do no good alone. The only plausible way I can think of to
deescalate slightly would be for Westmoreland to keep his mouth shut.
Whenever he makes a speech that now at last we are really winning the
VC start a new offensive. What a horrible thought that the two
candidates will probably be Johnson and Nixon.


/
/^ s^^














S mgof:
BB





SSeptember 27,-.16.

SSEP
Dear C ~ If you go t the !7
congress, you will see th GS /
o9g.f- ?- -I shall not r
need your book as a text in Pr
class but would be interested
to look through it. I am pretty D
good at picking up mistakes. St
I'll try your Chinese on S
Kuajita who is here this year an -
speaks English slightly better
than I speak Japanese. (But -
are even on oral understanding.iD
MOCKBa. KpeMnb.
DJnaroneutelcKHll co6op.
CeBepHan iacTb o6xoAHofl ranepeH. XVI B. q,
PocnHCh XVII II
qloro B. Po6uwHDa T


PR
-O


t-St


ofessor Kai Lai Chung
apartment of Mathemacs
anford University
anford
o-California


IOCTyflII lIljori3a. MocKlca. 1959 r.
0081359 17-973 Tl'n M 3 rica. 3. 231.
0000 U 75 K.








FAX 217-333-9576


L. L.
Secretary of Math Dept: Kindly pgone Prof. Emer. )oob and give him
the fax below. Prof. Chung at once

Doob: If you gave Catherine, Don, ...et al' my large HINT and they
^-L~M2 1.
still could not do it, they are really dumb. Here it is with more
clue. Hint: Given two orderings of N, it is trval to construct a
third one which Iran yoga o n .initial sectionstwith
either (both). Since the 'functions converge (a.e.) ~Rnll threejroute
their limns are same. (Three and half lines in MY style). FAX me
at (650)857-9532 to acknowledge receipt of HINT. Cheers. Chung
You can also phone me if you have franking pension right as I do,
but I am sending this at my expense to sade time.






'^at^ Settk '
vc 4i^^"











May 21, 1964
Dear Doob:

How many times must I ask you yo give me the complete title
of your pal Kawada's paper? Send it now.
I have just gohe over your minor corrections. Obviously you
did not read the MS, or otherwise you would not have wriiten
again "naturally progressively Borel separable Measurable". Please
read Def. 6 on p. 32. Why pile up adverbeswhen it is not necessary?
How would you define a naturally but not progressively Borel means. thing?

You are right about deleting "t hold-over from an earlier writing), but how about changing T to
T in (55)? Please check tgis. (You act like Michel in correcting
errors.)

I have no real objection to your-new pages except that it is aWbt
too terse. Can you write a little more like soie good probabilists As
Dynkin? In any case let me know when it becomes final so that I can
have these new pages typed up. I will add a footnote stating that
"right separability" and the corresponding theorem for the classical
case was proved by me (unpublished).

You should dash off a postcard to Meyer since in his book MS he
had first said that in our paper his theorem was proved with separa-
bilty thrown inA but he iight have changed his mind after having seen
our MS as there is not a word about it there\

I have no objection to sending the paper to TAMS, but we can also
consider Amer. J. of Math. as Hunt is an editor and he may read it
with a little more care than Michel\s gang there.

I think I will make corrections and some minor additions in my
,''e\y book withoutenlarging it. The royalty they are paying is now much
too low by American standards (it amounts ,to. 10 per cent or less net
while the standard one here is 15, quite a few 18). Please negotitiate
this seriously and let me know the answer.

My bounaary.theory is now really simple and complete (not in the
Feller sense but in the dictionary sense). To alleviate your feelings
about the complete failure of Martin boundary for this purpose* I shall
put in an:obituary note lamenting the cuteness but futility of the latter.
Incidentally I am amazed that you would communicate the Research Announce-
ment by the two Japs on the Martin Boundary which came to my notice only
recently. I had always thought these Announcements: were either for im-
portant things which could not await delay or brief interesting pearls.
I have not seen your article in the Atomic Sc.. (can you send one?) but
your guilt about Hirosima need not make you lose your math judgment.







UNIVERSITY OF ILLINOIS
DEPARTMENT OF MATHEMATICS
URBANA. ILLINOIS 61803


May 25, 1964
Dear Chung:
Kawada: 'On the measurable stochastic process'. (I hope Kawada's
English has improved since he figured out that title.)
(55) should be:

sBmx~BmmammarbmdenAmidmrmnjm (my God what a mess we have
made of that page; I just
noticed it.

((s,o): s
as you suggested. But then T should be T in (56), twice in (57),
a-
.the f.. at .-i. the isplaty ( .hali tC L No
I just realized that with your conventions that is correct.
How could I be expected to realize how unnatural your natural
definitions are? Why should not there be a process which is
naturally measurable (meaning the natural fields are those involved)
but not progressively measurable?
Unless you really want me to expand, I guess those new pages stand
as they are. When you have them typed send a copy to Meyer, so he
won't change his reference. I shall see him in about a month in Paris.
Amer. Journal is OK with me. Certainly Michel will not have our paper
reqd seriously.
Tell me what you really want out of Springer and I'll try to do
something. I can of course send a general complaint about low
royalties, but do you want me to demand a higher rate for your second
edition in particular? No, I suppose I do not need a letter from you.
I shall write simply that as it is at present the price of your book
is generally considered excessively high and its author considers its
royalty rate excessively low. Now he is considering revisions and
wonders what Springer's is prepared to do.
I rise in defense of my excolleague Martin to observe that if you
have found the Martin boundary insufficient, the converse is probably
also true.
My Bull At Sci letter had nothing to do with atoms or Hiroshima. It
was just carrying on our feud with NSF. Incidentally in the letter
from NSF to the AMS granting the money we want to MR, NSF makes detailed
demands on such things as the organization of prrofreading by MR which
shows how far NSF is willing to go in controlling math if we let NSF
get away with it.
Meyer's book ascribes the name 'prog. meas.) but not the idea to
us. I wonder whether that is a mistake by the translator.
Did I write you that Peter is getting married in August? You did
not beat him by much. He has known the girl for over 2 years and there
is nothing reasonable that can be done to stop the marriage, although
both he and the girl are ridiculously immature. She is graduating
from high school next week. He is just finishing his first year at
Knox College, and intends to be an actuary. We have agreed to continue
paying his tuition and in theory she will get a job to support them.
Actually my guess is that her parents will spoil them by sending
support.











7/14/1964
Dear Doob:
Are you sure that was the last revision needed? I won\t send
it until the paper is accepted, I am now into" the fifth chapter
of my book. Though this draft is very rough I am over the hump
and the book will be written. I want to keep it small say 300 pages
or less. As a sometime author maybe you can answer the following:
(1) Can you derive from Kol's 3-series theorem. that convergence'
in pr. of indep. r. v. implies a. e.? (without using ch. f.)

(2) Do you have a martingale type of proof of Levy's theorem
that if S is the nth sum of an essentially div. series of indep.
r. v. thar S >0 i.o. with pr. 0 or 1. (The general statement with
an arbitrart sequence A instead of 0 is trivially equiv. with this.)
There is a beautiful 2 Tine proof in the ident, dist. case (apparently
minded by Spitzer et al.)
Talking about Spitzer have you really read parts (f his book. He
tried too hard to avoid serious use of Markov chain theory in parti-
cular my taboo results, thereby-greatly obscuring some of the (admit-
ted less deep) arguments. For ex. he gave some foolish proofs (one
due to Wolfowitz) that any superreg. function on recurrent random
walk is a constant. The proof for M. C. is so much more transparent.
(It will be in my revised ed.) .o'r,


About Chacon everyone told me that the new Jacobs' lecture notes
on ergodic theory is good but I disagree. He did very exposition
though he wrote out at great length Chacon's later results.. Compared
to him Loeve must be regarded as-an outstanding expositor (I wish I
could tell Michel that, sincerely). C0acon of course thought it good
as the German prof practically licked his boots. By the way how come
your judgment about my pupil was so wrong? I never thought he was
too good, but always thought he was good enough. and I haven't had
to recant yet. Tle same applies to Austin* about whom you seemed also
to have to change youw mind. Would it not be better to be wise earlier'
If Brelot did what you said in driving and he had lived in Berkeley
can you explain to me why these guys don't die on the freeways? Even
the way I drive I always felt threatened there. How was Ito's talk
in Paris. He wanted to give me a huge MS with Nisio and I am glad
he took it back for Kyoto where no refereeing is necessary.







UNIVERSITY OF ILLINOIS
DEPARTMENT OF MATHEMATICS
URBANA. ILLINOIS 61803


May 10, 1964

Dear Chung:

I have considerably improved the last few pages, finally getting
separability, and even your right-separability tied in with the other
things. The argument used is a rather cute new one, the essential idea
being that if 4 is a function from T to T taking n only -contably
mary-val-e-s, and if the range space of the r.v. is compact metric
then there is a sequence (v.(t,w). such that lim 4 (t,w) exists for
all (t,). v. is integer-valued, increases with j,3and has reasonable
measurability properties. I think the idea should prove useful in
all separability arguments.
Corrections:
p. 3 would it not be clearer to add 'on T' to the first line of def. 1?
p. 6 (7) should be on the left J


p. 8 Add 'in (10)' to the last line of prop. 5.
p. 9 Replace 'any' by 'each' on line 1.
p. 36 (49) first a should be a. -e 3~ -/ -.' "-
p. 37 (52) is garbled: s (54) delete t p. 42 last line of prop:
modification of the process which is naturally progressively separably
Borel measurable.
Proof. The method of proof was suggested by P. A. Meyer.

(What I put before was to begrudging and sounds silly.)
pmm mnmmmmammghammaimmmamnid
pp. 44, 45 are enclosed. p. 46 is now the old p. 44 less the first
7 lines. Old p. 45 is to be thrown away and a new p. numbered 47 is
enclosed. The numbering of all pages in Section 5 should thus be
increased by 2.
What is the proper reference on the new p. 45 to your right
separability?
If you want to rewrite the material in any way I have no objection.
Note that I have omitted some proofs on new p. 47. I f you think they
should be included I can work them up.
I am still thinking a bit about this so do not send the material
off for a week, just in case. If you want me to send the paper off,
send me another copy. The Illinois Journal has a tremendous backlog,
over 18 months delay. Why not the Transactions, which is printing
extra issues to catch up?
Of course that plane density problem you sent a solution of follows
from Chung-Fuchs, but it surprised me anyway. My proof was more elementary
than yours.
You must have some psychological block about Kawada. FOR THE NTH
TIME: that reference is just exactly what I said it was, what you quoted
in your letter of May 1!!!






It seems to me that I have heard that the Air Force is willing to
give money to write books.
Who believes you have anything worth while on boundaries?
I have heard that someplace (Indiana?) is offering Rosenblatt
25000 but cannot believe it.
If your lectures are any good and are long enough I might persuade
Springer to stoop again. Have you seen Meyer's ms.? Tremendously
technical but very good.
Last AMS Council was very strong about not accepting agency imposed
members in our organizing committees. Albert was one of the strongest,
and in general he is hardly a foe of government money! The French are
sure it is politics because NSF would not give me money to go to the
potential theory colloquium this summer. Personally I sympathized
with NSF. Why should NSF pay for a French colloquium? If DeGaulle can
Sbuy his bombs he should buy his colloquia. The French were reduced
to paying for me themselves.
Did you see my letter to the Bull. At. Scientists (April)? I
have been getting fan mail, and even got a request for a reprint.




/h^)



















From the Preface:"In this monograph I present classical limit theorems
in the theory of probability and generalizations of these theorems, and
then I use a general method which permits even further generalizations.
The limit theorems are stated in such a form that they give connections
between the convergence in a certain norm, the so-called Gaussian norm,
of a convolution product and a corresponding sum, in the same way, that
there is a correspondence between the convergence of an infinite product
Tav, of numbers and the convergence of the infinite 'sum CL v-l] under
certain conditions. It would even have been possible to establish all
likmie theorems stated in this monograph as limit theorems for products
of abstract elements belonging to a set with properties which correspond
to some properties of the set of functions of bounded variation."
Thus, not only/sitribution probability functions butfou-.dtions of
bounded variation are considered, in the latter case howeverr they have
to be dominated by/monotone7bounded functions. Instead of the usual
method of Fourier transorm, a Gaussian transform is used although the
former is also used as in the discussion of infinitely divisible dis-
tributions. A modified Riemann-Stieltjes integral is used so that any
function of bounded variation cin be integrated.with respect to any
other. This requires rather technical handling to which the author
adds some unusual conventions, e. g. the symbol" 0C" wi.ich satisfies the
relAtions co+=-.o, co-= +o" and such that. (oo, o) as well as the
usual (-oo +oo) denotes the set of all real numbers. The exposition is
much detailed and presupposes only some elementary algebra and analysis
but the reader would have to be very determined to go through with it
all. The author syas. "Since I present a new method I need not quote
many earlier works on the subject. I shall not give any historical
background but refer to the well-known work of Gnedenko and Kolmogorocv,
Limt tkaaxzUa distributions for sums of jiadependent random variables
(1950)."










THE INSTITUTE FOR ADVANCED STUDY
PRINCETON, NEW JERSEY 08540




SCHOOL OF MATMATICS June 25, 1965
Dear Chung:
Finally am getting through packing and had time to look at your
question. Something is fishy somewhere. My reasoning is trivial. I
have not had a chance to look at Breiman, but note that your review of
his paper agrees with my description of den. atomic. Take C as satisfying
this definition. Then Bl is true so thereis positive probability that
paths hit C i.o. so the set of fine limit points of C on the boundary has
strictly positive harmonic measure. Now let a be an ordinary limit point
of C on the Martin boundary and let A be a sequence in C converging to
a. Then almost every path meeting C i.o. meets A i.o. and so must
converge to a (since a.e. path converges). But then C only has one limit
point on the boundary, and this must have strictly positive harmonic measure.
Finally, if a fine neighborhood of a does not contain some infinite
subset of C, say A, a.e. path hitting C ihitm i.o. also hits A i.o. but
then A has a as fine limit point so it must meet every fine neighborhood,
contrary to hypothesis,. So if our version of Breiman is correct my
description in terms of fine neighborhoods is correct. On the other
hand it contradicts what I say is his equivalent condition your review
states it differently but the two seem to come to the same thing. Your
example is one which obviously does not satisfy the definition so the
state space is not denumerably atomic.
NO! I take it back. Since we do not know what the subregular functions
are in your example, it is not at all clear that there is any contradiction.
What is the fuss about? '.hy do you think your example is one with a
den. atomic state space?????
Elsie is in Europe now, and Deborah and I leave next Tuesday to
meet her in Budapest, where we spend a few days with her brother. Then
we rent a car in Vienna and drive around looking at scenery in Austria,
Switzerland, and southern Germany. Letter to me c/o
American Express
6 Stadelheimerstrasse
MIunich
will reach me (with some delay).
How did Brelot work out? Are you going to hoscow? I shall probably
not go, because the trip does not seem worth the money, and there is
nothing I can think of to buy for the rubles in my bank account there.
Have a good time at the Berkeley Symposium. (Ha!)



.^Wh-








THE INSTITUTE FOR ADVANCED STUDY
PRINCETON, NEW JERSEY 08540




SCHOOL OF MATHEMATICS February 18, 1972
Dear Chung:
Hope that the f enclosed will be effective. What you and i
Sam should do is play some probabilistic game: winner gets the man
he wants.
Surely there are problems for your book that are relevant.
Sometimes the classical ones just need an interpretation to oe so.
For instance a test to see whether vaccination is effective, or
an examination of a poll: is a difference of x percent between
Kennedy and Nixon significant. Or an examination of the evidence
that makes anyone think that Brownian motion is what mathematicians
say it is. Or a risk problem a la the Swedes who seem to use this
stuff in insurance (made into discrete time of course). Even the
first boundary value problem for discrete harmonic functions on a
net can be made pretty trivial and is apparently used practically.
I admit it is a lot easier to do math than to look up such stuff.
Similarly I remember how I resented checking up historical points
for my book (and did not do a very good job).
It is possible Bill Stout would be interested in collaborating
on your book. He is very hardworking, reasonably talented, somewhat
pedestrian. He has written a draft of a monograph on certain aspects
of martingales: there is nothing elegant about what he does but he
is certainly competent. He is assistant prof at Urbana. You may have
seen things he did on stopping times, and Meyer refers to him for
something or other about iterating logs.
I vaguely remember having once seen the McKean Newton-Gauss
paper but could not find it when I looked it up in the Kyoto
math j. here. Give me the complete reference, so I can enjoy seeing
my name in print, and I might even read it. I always find Ito-Mc a
challenge, sometimes amusing.
Here is a funny problem I have been working on, without much
success. Let D be the unit disk. Then there are various theorems
that say that at a.e. point of the boundary if one approaches the
point along a suitable filter a bounded analytic function has a limit.
Example a) approach along radii b) nontangentially c) fine top.
It is classical that if there is a limit along a radius there is even
a nontangential limit, so A) and b) are really equivalent so even
though b) looks better, it isn't. On the other hand c) is actually
better because c) implies b) but not conversely. Now is there a best?
Best would mean that the approach filter should be minimal, that is
its sets should be as large as possible. It is not too hard to make
this concept rigorous, but there are too many sets of measure 0 for
Zorn's lemma to be used to get maximal methods of approach (or minimal
thicheverx it is I want). The best way to analyze it is to take the
compactification of D allowing every f a continuous expansion (max ideal
space of the algebra of bounded analytic functions.) Each point of the
boundary describes a way of approaching the boundary and the trick is
for each point of the usual D boundary to maximize the set of points
of the new boundary 'over' Q with the property that on approach to
that set f has a limit for a.e. Q. Too bad I have few results do far,
but I like the problem.
.af!-~--^lsa^*^









THE INSTITUTE FOR ADVANCED STUDY
PRINCETON, NEW JERSEY 08540



March 3, 1972
SCHOOL OF MATHEMATICS

Dear Chung:
As far as I am concerned, your treatment is not covered
by McKean, who in any case mumbles about smooth manifolds. But
I object to your first sentence. For a while I thought you
really only meant to have a single process in my sense. Of course
you are assuming a variation of Hunt's hypotheses and I think you
should say so. Why imitate McKean's vagueness? Of course you
will have to insert references.
I suppose that you have received the long Pittenger-Shih
preprint. Is that length really needed for their work? I wqs
amused to see that the compactification of the countable state
space of a chain is called the Doob-Ray compactification. I
'arrived pretty late in that scene, but perhaps the reference is
just an indication that Pittenger still has no job, and is
being cautious. Walsh now has 2.5 job offers, he writes me,
and does not know which to accept.
From Sam: Many thanks for your letter on Walsh. It is
a very strong letter which is rare for you and I am very impressed.
Believe me I will try very hard, as I have always, to support
Chung's candidates and I will definitely arrive at a consensus
with Chung in order to get a young probabilist at Stanford, which
is sorely needed.
Have you seen a recent report on math in Israwl? Sam is
listed as 'formerly of Stanford'.
I object to writing vaguely on page m+n with the justification
that there was care on page m. Where does that leave a reader
who either does not remember page m or is consulting the book
and has not read m or read it long ago? I did not mean to imply
that I understand Feller (or anyone else) on genetics but not you.
I have never read any account I could understand. Why not see
what hypotheses you really use instead of throwing around words
like everyone else? The key is to define random mating I suppose.
Just define some model that makes your calculations work and
say that the real situation is identified, for better or worse,
with the model. I have not checked, but I think Feller has that
rearrangement of terms OK in his expectations. I think I remember
calling it to his attention.
Jack's heart attack was very mild. He is already out of
the hospital. Are you claiming credit for predicting it?
Here is the kind of example I mean. I use it when I teach.
Elsie's old text describes blood counts and gives the procedure
a lab technician is to use: count so and so many squares etc.
One can easily calculate the distribution of cells per square
and see if in fact the technician should get about the same result
if the count is repeated. To find a proper vaccination example
you would have to look in a public health book and see some basic
data used to justify vaccination. Or look up 'vaccination' in a
library catalog. I admit that sort of research is a lot of work,







To find other examples look in Biometrika or applied journals of
various sorts. Journals on quality control might have something
from actual experience. During the war, ships were run over
magnetic installations and I heard these were designed by difference
equations, in other words harmonic functions on a lattice. (The
problem was to make the ships safe against magnetic mines.) In
other words there are really infinitely many examples around, if
only one wants to spend infinitely much time digging them up.
I wandered through the Princeton store yesterday and was amazed
at the number of probability books there, some tith statistics,
but lots alone, several with the words 'Stochastic Processes'
in the titles. How are you going to make yours distinctive?
It will be more nearly correct and therefore duller than many.
It will not specialize in some particular topic as some do to
get a talking point.
On insurance risk: Cramer etc. use criteria to find the
probability that the company will go broke in n years. Some of
that could probably be simplified for your book. As I remember
the fundamental inequality is a simple martingale one. Of course
you would use discrete time.
I am still tied up with Gelfand representations and
analytic functions, and making no progress. The general problem
is this. Let f be a bounded analytic function in the unit disc, D.
a) LOCAL problem. Let A be a filter of subsets of D, converging to 1.
Is there a coarser fileer B independent of f such that whenever
an f has a limit along A then f even has a limit along B! For
example does a limit along nontangential approach imply one along
a better approach? (The answer to that is yes, but not much better.)
b) GLOBAL problem. Let A be a filter of sets converging to 1, as
above, and let A be the rotated filter converging to z. For
some choices of A, any f has a limit along A at z for almost all z.
This is of course Fatou if A is nontangential approach. Is there
a best, that is coarsest, A? I don't know, but nontangential
can be improved for example by combining it with fine topology.
To handle it I compactify to get all f's to have continuous
extensions. I have lots of unsolved problems but no decent
methods so far. The general problem arises in lots of contexts.






(L-- v









THE INSTITUTE FOR ADVANCED STUDY
PRINCETON, NEW JERSEY 08540




SCHOOL OF MATHEMATICS
March 16, 1972

Dear Chung:
I forgot to ask you in my letter if you realized how trivial
your result is-in the discrete potential theory case. (This is really
Hunt. )
Define g(i,j) = Zp .(n) -and let f(i) be the probability that
o ij
x ever (n>O) hits the transient set A, starting from i. This is
the equilibrium potential and

f = f-pf +(pf-p2f) + ... = g(I-pf).
That is, f is the potential of h=(-p)f. Here h(i) is the probability
starting from i that x never (n>O) hits A. Then

P.{last hit of A is at j} = Z 2 p..(n) h(j) = Z g(i,j)h(j)
A n A

which is your theorem, in this trivial came. Note that to get the
representation of f I used the fact that
n >0
pnf ;;> 0
which is L1 convergence. This is essential.
Actually the real reason I am bringing this up is that in
discussing first hitting timws and harmonic measure the key fact is
that a bounded harmonic function gives a martingale on paths.
Therefore I feel that the right martingale in the present situation
must be theg key to the connection between last hitting times and
equilibrium measure! Do you believe this?



t~j^^^









THE INSTITUTE FOR ADVANCED STUDY
PRINCETON, NEW JERSEY 08540




SCHOOL OF MATHEMATICS
April 12, 1972

Dear Chung:

Just returned from a tour of Canada. I guess you were
there not so long ago too. I was interested to see the number
of European mathematicians around Ottawa. Now that jobs are
hard to get, I suspect that this will cause trouble, and in
fact I heard a few rumbles from Canadians.

I am game to cooperate in writing about gems. You
misunderstand that convergence axiom. In axiomatic potential
theory there are various convergence axioms, but they are for
sequences of harmonic functions in each context, monotone sequences
of continuous functions. If the context is elliptic the axiom
is that the limit is harmonic, if finite, but third does not
seem appropriate in the parabolic case, for example for 'harmonic'
meaning a solution of the heat equation, and something like my
axiom (that the limit is 'harmonic' if finite on a dense set)
is used. This result or hypothesis does not involve anything as
deep as the theorem about a decreasing sequence of excessive
functions.

I object to the following in your equilibrium measure
business. There ought to be one theorem involving the first
hitting of a set: going forwards this should give harmonic
measure etc.; going backward it should give equilibrium measure,
in the right context. How about it?

Do you have any definite 'gem' proposals?




A--









THE INSTITUTE FOR ADVANCED STUDY
PRINCETON, NEW JERSEY 08540





SCHOOL OF MATHEMATICS
April 18, 1972

Dear Chung:
Brelot proved that (classical potential theory) the set
of points of a set at which the set is thin is polar, in
particular that a set which is thin at every point is polar.
This was done in the early days of the fine topology, 1944,5.
I have seen your rroof someplace cr other I am sure,
although perhaps the proof was in a different context.
The idea that the probability of hitting a set from x is
the equilibrium potential must be pretty old. Since everyone
knew about the relation between partial diff equations and
Brownian motion, from Fokker-Planck, lots of people must have
solved the obvious Dirichlet problem to get the equilibrium
potential, at least for a smooth set, noticing the probability
interpretation on the way. I haven't got my old subharmonic
f. and rot. theory handy at the moment so cannot say whether
or not I stated the inter-retation for all compact sets, but
I think I did. [I'll bet anything that Levy among others used
the fact to find the probability of hitting a set by solving
the p.d.e.]
I still think that there must be a unified way of looking
at first hitting times forward and backward, but am so tied
up with my damn function algebras and boundary limits that I
have not been working on it. How about it?

/









THE INSTITUTE FOR ADVANCED STUDY
PRINCETON, NEW JERSEY 08540


May 5, 1972
mamnffhmngnm
SCHOOL OF MATHEMATICS

Dear Chung:
Here is another proof of your result. I give it for the classical
case, and have not checked to see how it would go over to Hunt processes.
Let A be a set and Let L(x,.) be the distribution of last hits by TA
Brownian paths from x. Then L(.,-4) is the equilibrium potential.
The proof that this function is superharmonic shows more generally
that L(.,B) is superharmonic, harmonic off the closure of B. Take A
compact. Then if B is compact L(.,B) is a potential (because it is
dominated by the equilibrium potential) and is harmonic off B, so

(1) L(x,B) = fg(x,y) m(B,dy)
where m is supported by B, and is uniquely determined. Let B be compact,
not meeting B. Then since L is additive

m(B UB o dy) = m(B,dy) + m(Bo,dy)

= m(B,dy) on B
When B increases to the complement of B we get

f m(A,dy) = m(B,dy) on B.

The left side is of course the equilibrium measure, so (1) is just what
was to be proved. Ofcourse this is less probabilistic than your
approach. I have not tried approximating to allow A to be Jorel or
worse. My proof is so easy that I doubt that it is new.

Regards,









THE INSTITUTE FOR ADVANCED STUDY
PRINCETON, NEW JERSEY 08540





SCHOOL OF MATHEMATICS
May 16, 1972
Dear Chung:
In classical potential theory a positive superharmonic
function is a (Green) potential if and only if it goes to O at the
boundary 'sufficiently fast'. The latter can be expressed in various
ways, and means more than that the function u has limit 0 along Brownian
paths or equivalently has fine limit 0 almost everywhere on the boundary
in terms of harmonic measure. One wqy to state the condition is the
following: let R C R ... be an increasing sequence of domains, relatively
compact in the domain of u, increasing to the domain, and let T be the
n
first time a Brownian path hits the boundary of R from some fixed
point in R Then the condition for a potential is that E{u[x(T )]}-> 0.
Then any positive superharmonic function dominated by a potential is a
potential. In fact in everybody's axiomatic potential theory the
definition of a potential is that it is a positive superharmonic function
which dominates no positive harmonic function except the function O. Of
course this also implies that a superharmonic function dominated by a
potential is also a potential. Corresponding definitions are made in
discussing supermartingales: every time the word 'potential' is used
there is a condition making an L limit 0. (Actually Meyer's first draft
of his book did not have this, and I called to his attention the fact
that it is needed.) Similarly in discrete potential theory, if you
look at my paper, or Hunt's. I don't understand the first paragraph
of your letter: to everyone potential means the result of operating
with a kernel on a function or measure; sometimes a potential is defined
otherwise but it then becomes an immediate problem to find a kernel
to match so potentials have the usual meaning.
We plan to go to Europe for about 6 weeks, starting some time
in June, but have made no definite plans. I may go on a canoe trip
in Canada the first half of June. At first I thought we should go to
Moscow and spend some of my banked rubles but it turns out that the
rubles cannot be used for air passage there or for hotel or food there
so I gave up and we shall not go there. We'll pro:.,b-,b rc:ilt c:r and
:mostly : .ncder .round central Europe.




^L^-^^/






UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801



April 25, 1974

Dear Chung:

I like your predictability proof except what is that

gobbledegook on the top of p. 4? I can't make head or tail of

it. Isn't the following true? X_ is measurable T_ so if there

is continuity at T it is true (but not quite trivial) that 5T =9T.

Why don't you say this instead of the vague allusions?

What is the proper reference for your 'useful fact' on

p. 2? The only one I know is Mertens. I have been doing some

slightly related things. First: Do you know this (rather trivial)

result. Most people deny it:
A1n ; x measurable 5 ; x -> x a.e. ; x and x integrable.

limammmmEwmmm6ma m m m mamLmamlmmmmmamupmamu m

Then E({x-xn 1) -> 0 a.e. (but not necessarily L1).

I use this in the following. Let X,?. be an adapted process on

[O,a ) and suppose that 9. is right continuous and that all the

algebras contain the null sets. Suppose that x(T) is integrable

for every finite stopping time T, that X is right continuous, and

that z is any integrable random variable. Define y(t) = E(Iz-x(t)Jl, t).

Then this definition can be made in such a way that Y is a right

continuous process, with left limits if X has left limits. Y is the

process everyone uses in discussing BMO but I haven't seen this property

stated.

I have always thought that Neyman started the Bekkeley

school and thereby put US statistics on its feet. Don't you agree? The

only objection I have had to his statistics is some stuff he did during

the war in which he calculated the probability of a bomb sinking a
destroyer, several decimal places and his hypotheses were quite absurd.






/
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801






October 9, 1974

Bear Chung:

What is all the fuss about? Let x(.) have independent

increments on [O,o ). And let A be a set in the tail field. Then
B
it is independent of every set depending on differences because

such a B can be approximated arbitrarily closely by a set depending

on differences in some interval [O,b]. (This does not use

martingale theory, just the definition of what a set depending

on differences is.). Hence A is independent of itself so has

measure 0 or 1. This is exactly the same as one of the Owl law

proofs in the discrete parameter case. Your example is no

good because if a prodess is identically z(w) it has independent

increments all right but all the increments are 02!! I don't know

what your other example is supposed to prove. At any rate as

you can see from my proof above stationarity has nothing to do

with the problem.

What a coincidence. I am also to give a BM course

the second semester but have done no planning for it as yet.

For tied down Brownian motion I always think of the finite dimensional

distributions as obtained explicitly (trivially) using the density

and then point out that this is an honest process by Kolmogorov,

and in fact that this is just a conditional h-process so has

continuous sample functions and can be interpreted as conditional

B.M. I never looked at his Theorem 42.5 but I suppose I'll have

to in a few months.

/- <: 7 ^

4 _^^ ~^ '







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA. ILLINOIS 61801


October 23, 1974

Dear Chung:

You baffle me. I still don't know what all the fuss is about.

What you are after is a special case of Hewitt Savage in the continuous

parameter case. The following is a translation into the c.p. case of

the standard proof of H-S in the discrete parameter case. Let x(.)

have stationary independent increments with x(0)=0, and suppose first

that the process is a coordinate process in function space. (That for

your nonsense about right continuity!) If a>O and 6>0 and t<6

replace x(a+t) by x(a)+x(c+t)-x(c) and

replace x(c+t) by x(c)+x(a+t)-x(a) getting in this way a measure

preserving transformation of function space WHICH DOES NOT CHANGE x(s)

FOR LARGE s. Now suppose A is in the tail field. Then if s>0 there

is a set A defined in terms of t ,...t with d(A,A ) & n &

function as usual). Suppose t. K for all j, and interchange the
j-

intervals (O,K), (K,2K) as just described. This measure preserving

transformation leaves A invariant and A = 1( (tl),...,x(t )] B}

goes into A' = ([x(K+t )-x(K),...] (- B). Hence A and A' are

independent. Moreover A is invariant so d(A',A) < so d(A ,A')<2s

and this contradicts independence if e is small, as usual, or better

P(AnA') = PA P(APA') becomes P(A) =P(A)2 when e ->0. For a general

process, not necessarily of function space type, the result is immediate

by the usual map. Right continuity and such have nothing to do with the

theorem.

If you write up your Brownian motion send me a copy. My book is

at the following stage. I have pencil scribbles which I am now typing





to get a first rough draft. By the end of the week about I shall have

this draft of classical potential theory like Helms' book but much

better organized in my opinion covering the usual stuff of properties

of superharmonic functions, the Riesz decomposition, various maximum
Dirichlet problem
principles, capacity and polar sets, fundamental convergence theorem.

I also have typed but have not checked at all the section on martingales

and as much Markov processes as I shall need. (I do progressive measurable

things but nothing worse, but prove right continuity of sigma algebra

families and strong Markov under reasonable conditions. I Show that

hitting probabilities of analytic set-s are independent of the process

involved, just depend on transition probability and initial distribution.)

I have scribbles on Brownian motion and some on the interrelations with

probability.,but have not started typing these yet. So all I can send

you is the classical potential theory, which you probably don't have any

use for. It is, 100 pp. of classical material with no probability and

no BM. I only have one carbon so will Xerox a copy if you really have

use for it.

I have not heard from Guggenheim about you yet. You are way ahead

of me in publication. NO! I typed that before counting (I have an index

card for each paper) I have 83! Of courses I have a head start of quite

a few years. [The number I am proud of is considerably smaller.]


; ^ ^/






UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801

January 1, 1975

Dear Chung:

I have not worked on the probability that a BM from 0 reversed

at time t will hit x before 0, but my instinct would be to work on the

probability that a BM path from 0 which las value y at time t (tied

BM) reversed hits x before O. If this is known the answer can be

integrated over the distribution of y. Now the value is the same as

the probability that a BM from y which is at 0 at time t gets to x

before O. In other words last hits are no different from first ones.

Looked at in this way it would not surprise me if Bachelier had calculated

it. He calculated a lot of complicated things. I'll bet a nickel or so

that your approach comes to about the same thing as the tied BM approach.

I am trying to avoid doing too much general process work in

my book so don't tempt me. The book is supposed to be classical potential

theory and BM and I am only putting in general ideas for orientation

as needed. For instance the word SEPARABILITY is not in the book.

Your treatment of increasing processes is the obvious one

except that you assume Borel measurability of the processes but I don't

see where you use it. Actually it is not hard to show that you might as

well assume right continuity. In fact x(.) is a jump function iff

g&g x(t+) which is a well defined random variable by your hypothesis iW

b. A, jump functions and if two of your processes have the same finite

dimensional distributions the corresponding right continuous processes

also have the same finite dimensional distributions. (Of course you then

could have progressive measurability and use a fancy proof of your desired

result.)

I am hoping to be able to avoid going into this much detail in

my book, but I'll see what I need as I go along.






UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA. ILLINOIS 61801



February 26, 1975
Dear Chung:

Enclosed is my proof that a superharmonic function is excessive

(if positive). More exactly I prove a supermartingale theorem which

gives the inequality for excessive functions. Later I prove that

a superharmonic function is continuous on Brownian paths, which

implies by Fatou's lemma the limiting equality part of the definition

of an excessive function.

In the other direction, the main thing is to decide on how much

of the general theory is to be used. For instance if one can assume

that an excessive function is continuous on Brownian paths then the

function is superharmonic or identically +a0 by the following argument.

Since uAn is excessive if u is and since the limit of an increasing

sequence of superharmonic functions is +oo or superharmonic it is

enough to prove that a bounded excessive function is superharmonic.

But such a function u is a superkartingale on Brownian paths from a

point. The supermartingale selection theorem gives that u satisfies

the average inequality for superharmonic functions. The tricky part is

to prove that u is lower semicontinuous, but since you only seem to be

worried about integrability I'll omit further proof. In fact I have

no treatment as yet I have liked enough to write up. What I may do

is to use the fact that the Brownian transition density for a domain

is nice and continuous. (I have a reorganized and reasonable treatment

of Hunt on this density, involving conditional B. motion and parabolic

functions.)

Maybe the problem with Ito-Mck is that 'generator' is nowhere

defined! Just glancing through I saw no definition! I have not come

to barriers yet.







2?



11. Superhermonic functions composed with Brownian motion. Let

D be an open subset of E In the following lemma, Brownian motions

(x (.)," (.)) will be used, with initial distribution p supported by D.

If i:- su-r-orted by the point F of D the subscript F will be used instead

of p. It will be convenient to suppose that in each case 5 (.) has been

chosen to be right continuous, with each Y(t) containing all the null

sets. Such a choice is p'o;sib.e according to Section 6.

Lemma Let m u be positive and superharmonic on D, let S

be a point of P and let T be the hitting time of bD by the Brownian

motion {x(. ),J (.). Then the process (u[x(t)]lT >t, (t), t 0) is

a supermartingale, except that the parameter value 0 is to be omittec1

if u(F) = o .

In the following '-roof B(.,s.). is. the. ba1l with center T, in D

and radius EA(IT,-bD(/2),.. Let p. be a probabilitydistribution supported

By D and define

To =0, T =inf(t>T: x (t) (- B(x (T ),e)}, n>O.

The function T1 is optional for 5 (.) because the vector process
N N
([x (.),x (0)],0 (.)) is a continuous process with state space E X E

and T1 is the hitting time for this process of a closed set in the state

space. The function T2-T1 plays the same role for the Brownian motion

(x (T +.), (T +.)} that T1 does for x (.) because the latter process has

the strong Markov property. Hence T2-T1 is optional for 9 (Tl+.) and

therefore (Prob. Section 11.6) T2 is optional for 9 (.). Proceeding by

induction every T is optional.for (.) and in view of the strong Markov

property the discrete parameter proess,{x (T, (T), n>O is a
1, 111 1(Ti), n>0}
Markov process with state space D and stationary transition function

the uniform probability distribution q(T,.) on BB(r,c). Since








71'Bc ,^ 2t



q(T,u) = L(uIhi) ) u(T) for T in D, the process

(u[x (T ), f (T ), n>O) is a supermartingale, if p is chosen, as .
p so
is supposed from now on, that E(u[x (0)]) n-->an
is the hitting time of bD. Choose t >0 and define

k = min(n: T >t) if T >t
n -a
= oa otherwise.

Then k is optional for J (T.) and lim T = t a.e. where T > t.
E: e-> k CD
e-->0

If u[x (T )] is defined (with some violence to notational conventions)

as O, the process (u[x (0)], u[x (T ))] is a supermartingale (Prob.

Theorem III.14) so

(11.1) Efu[x (0)]} > E(u[x (T )]}

and this inequality becomes, when E-> o using the lower semicontinuity

of u and Fatou's lemma,

(11.2) E(u[x (0)]} > E(u[x(t) 1T>t)

If u() )
inequality ] for the process of the lemma; if u(T) = c the inequality

is trivial. Hence the process of the lemma is a supermartingale over

the parameter range for which the random variables have finite expectations.

If u(C)
nothing more to prove unless u() =o In the latter case let B be a

ball with center 0 and closure in D, and define u' =(~7BU Then u' is

positive and Eperharmonic on D and u'(e) is finite so E{u'[x(t)]) is

finite for all t. To prove that E(u[3(t)] 1T>} 0 it need

only be shown that E(u[x (t)]; x(t) B)
u is jh integrable on B.







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA. ILLINOIS 61801



April 17, 1975

Dear Chung:

Enclosed may interest you.

I havd to write a review of your elementary book for the

American Scientist just a perfunctory one of a few lines, but took

the occasion to have another look at it. Luckily it will not be a

real review because I have some objections! Here are a few things

you might fix in a second printing:

p. 87 (4.4.3) T should be X.

p. 98 Line -7 be should be the.

p. 195 Line -8 fix spelling of wheel.

p. 196 Contrary to your calculation, 11137.3. Fix that decimal!

There is confusion frequently on whether 'probability' is

a mathematical subject or a description of reality. For example oh

p. 25 I would like to have seen a remark that the mathematician can

define probabilities as he pleases, but the question is what assignment

yields a proper model of tossing a pair of coins.

p. 162 Line -3 is a mystery.

p. 178 TSK' On line 4 you restrict validity to. Izl <1. A little farther

down you evaluate g'(l)' And then you have the nerve on the next page

to refer to a 'terrible proof'. And why don't you say that the generating

function is Ez X) ? That yould make Theorem 6 trivial and convolutions

unnecessary.

p. 232 is enough to make a statistician's hair turn white. What does

it mean to say 'we can be 950 sure that...' Inverse probability, Bayes

or what? This is a standard controversy.

p. 233 Since it is not 'logically possible' to toss a coin indefinitely







unless you are coeternal with God, just what does it mean when you say

that something 'ma 'almost never happens'? Even though you say it in

italics it ought to make sense. (This is the result of never being

clear on the relation between mathematics and real life.) This

reminds me of how Ville solved the problem of whether in fact for a

balanced coin the success ration had limit 1/2. He wrote that since noone

could toss a coin infinitely often it was OK to say there was limit

1/2 because the assertion could not be contradicted. And he, like you,

thought he was making a contribution with such nonsense.

Yours in grief,










/L-6 L
ri- .^ ^ y~~r *'*^" -^"wC y







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801

October 31, 1977

Dear Chung:

I guess that long book is not long enough. If f=Ptf then by

hypothesis Ptjfj = ff(y)l exp[ly-xl2/2t] dy t-d/2 < oo.

That integral is formally parabolic (= satisfies the heat equation)

and actually by Fubini it has the average property of a parabolic

function. Since it is finite valued it is parabolic and therefore

infinitely differentiable. Then f, majorized in absolute value by

PtIfJ, is locally bounded. Let fo(x) be the average of f on a ball of

radius 1 center x. Then f is continuous and (Fubini) f = Ptf .
o o to
Thus f on Brownian motion is a continuous martingale and stopping
0
at the boundary of a ball gives the harmonic function average property.

This makes f harmonic. If radius 6 instead of 1 is used and 6-> 0
o
we get a family of harmonic functions, locally bounded (do the same

averaging to Ifl) and converging a.e. to f. Hence f coincides a.e.

with a harmonic function g. Since Ptf = Ptg and since the harmonic

function average property gives Ptg = g, f = g. That is probably not

the easiest proof but it is the first one I thought of. Probably it

could be done in a more elementary way. Yes: the key fact that f is

locally integrable follows at once from finiteness of Ptlfj and this

is all thht is needed to form those ball averages. Parabolic functions

are not needed; ignore the first part of the above proof. Probably

the rest can be simplified too.

Give my love to the mandarins. Did you see that Salisbury writes

that the Chinese are now wondering whether there was a gang of 5,

including Mao?!,I

----U,---







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801


December 8, 1977

Dear Chung:

I feel that you don't understand what the real issue is in comparing

the various proofs of the fundamental convergence theorem, regularity

of boundary points, and so on. The hardest organizational problem I

have had with my book was determining the order of the development.

My solution has been to first write a fairly complete version of

nonprobabilistic classical potential theory, a sophisticated version

of Helms' book. Next I have probability: general stuff on martingales

Markov processes, analytic sets,... leading into Brownian motion.

Finally the third part of the book combines the two. The separation is

rather artificial, and I keep finding that it is more natural to put

things from the third part into the first. Even the first part presents

great difficulties in order. One can try to get the Dirichlet problem

as soon as possible. This seems to be the best way of handling general

axiomatic potential theory. But what I do is to try to get the fundamental

convergence theorem as soon as possible, and delay the Dirichlet problem

until late, to be able to use polar sets and so on when I come to the

D problem. The moral of all this is that there are 'trivial' proofs of

the various basic theorems which are quite incommensurable, because what

is trivial in one order is impossible in another. For example here is

the proof I shall probably use that a point z is a regular boundary
fine
point if and only if z is a limit of the complement of the given domain D.

Let B be the ball of radius r, center z. If Z is regular define u as
r r
the smoothed reduction of 1 on ( N-D)ABr relative to B1. Then it is

almost trivial that the restriction to DAB1 of ur is the Dirichlet








solution on this set for the boundary function equal to 1 on cD and 0

at the other boundary points. By regularity u therefore has limit 1

at z along D, and on the other hand the reduction u is 1 quasi everywhere
r
on (E -D)nB1 so it is trivial that u has fine limit 1 at z and so is 1
1L r
there. Since this is true for every r the set (EN-D) is not thin at z.
N
Conversely if E -D is not thin at z define 4(x) =l-Ix-zI on B and let u
smoothed 1
be the reduction of 4 on (EN-D)nB1 relative to B1. Then u is superharmonic

on B1, with value 1 at z by the nonthinness hypothesis, and u is

continuous at z so 1-u on DhB1 is a barrier and z is a regular point.

Now this proof is very easy but only because all sorts of background

is available. Similarly the proof Brelot gives (Springer LN 175 p. 34)

of the fundamental convergence theorem is very trivial and very elementary

if one has defined semipolar to match and your proof is absurdly complicated

and deep in comparison, as is mine by means of crossings.

All this shows that Mao was wrong: it is sometimes better to start

a trip with 1000 steps.






On looking over what I have in usable form in my book I see that all I

really have is most of the first part (classical potential theory) and

scattered parts of the second, so I guess I don't have what you need.







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA. ILLINOIS 61801

March 29, 1978

Dear Chung:

Consider the following three classes:

(a) Differences between pairs of positive harmonic functions

on some specified connected set. (Let h be a strictly positive element

in this class, the reference element.)

(b) Finit4 signed measures on a specified measure space. (Let

X be a smtahmi positive (A 0) element in this class, the reference

element.)

(c) Differences between pairs of positive martingales relative

to a specified parameter set and a algebra family. (To make life easy

take as thereference element the martingale =1.)

All three classes are vector lattices in the obvious order, and

the usual lattice definitions are made. For example a positive harmonic

function is 'quasibounded' if it is the limit of an increasing sequence

of positive bounded harmonic functions relative to h (that is

0
mzK1itzxuzk if it is the limit of an increasing sequence of positive

measures bounded relative to X (that is OnT, n Jn Cn ). The

corresponding class (c) is the class of uniformly integrable martingales.

A positive harmonic function u is 'singular' relative to h, that is u/h

is a singular h-harmonic function, if u/h majorizes no bounded positive

bounded h-harmonic function except 0 (u>v with 0
A positive measure p is singular relative to X if it majorizes no

bounded (relative to X) measure other than O, that is p.>v with

0
Suppose that somehow a 1-1 correspondence u <> M has been set
u







up between the elements of (a) and (b) which is linear and order preserving,

and makes reference elements correspond to each other: M =X. Then
abs. cont.
quasibounded [singular] elements in (a) must correspond to quasibounded

[singular] elements in (b), so that the decomposition

(1) u = u + u (u>O)

(in which u/h, u /h, uqb/h are positive h-harmonic, singular h-harmonic,

quasibounded h-harmonic) corresponds to

(2) M = M + M
u u u .
s qb

in the sense that (2) is the Lebesgue decomposition of M In (a)
u
if the domain of the functions is assigned its Martin boundary the

Martin representation sets up just such a correspondence, in which

u = -K(T,.) Mu(dT), where K is the Martin kernel. This correspondence

has the nice property that u/h has the BMh limit dMu/dMh a.s. so in

particular if (and only if) u/h is singular the limit is O. A direct

proof that the singular case gives limit 0 is easy as follows: By

the supermartingale inequality if u is positive

(3) u() > E{z ) where h is the BMh limit of u/h from the
h
initial point (.. The right hand side defines an h-harmonic function

of g, majorized by u/h and E(z A c) defines a bounded h-harmonic

function of g majorized by u/h and so is 0 in the singular case.

(In my book I carry this business through with lattices and allow

in (a) differences between positive superharmonic functions, with

corresponding generalizations of (c). If (c) is a positive potential

quasiboundedness means the process has a Meyer Amzmm representation.!To

orient readers I have an appendix on lattice theory. It was HELL working

this appendix up.) In the general case if u/h is positive but not

necessarily singular the right side of (3) defines the quasibounded

component of the left side and the difference is the singular component







h
which does not contribute to z .

As you can imagine I haven't the slightest idea when I first

heard of singular harmonic functions. To my surprise now on looking

back at my confused and confusing papers I find I never used the

word 'singular' at all. It was clear from the start that the difference

between right and left sides of (3) had BMh limit 0 but I don't remember

when I first knew that the function should be called 'singular'.

According to Brelot's work the Dirichlet solution for h-harmonic

functions on the Martin space, for the boundary function f, is given

by

S= fK(T,.) f(T) Mh(dT)/h so Mu is absolutely continuous

relative to Mh with fdMh = dMu, and my Fatou theorems were always based

on the result that f is the BM limit of u/h. Thus it was always

clear that what I proved wqs that u/h has BMh limit dM /dMh. I admit

that to my amazement I can't find a place where I wrote this although

I swear I 'remember' writing it. Probably what I remember is stating

this in innumerable talks and in courses! Lamb of course knew it too

from my course or conversation and 'well known' is not unreasonable.

I forgot to ask you about the China trip. Elsie may wangle a trip

there with a Planned Parenthood group and thereby see what she wants

much better than going with mathematicians.

How are your eyes? I have entered the club, having had my left eye

worked on (cryosurgery) last week because the retina was slightly

detached. I was getting funny light flashes in that eye.



//\i 1 -~-~
/^ ^







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801

July 13, 1978

Dear Chung:
Of course I'm retired, and it is great! (At 80 percent of my pay.)

(Incidentally do you still prefer double space typing?) I have decided

to take retirement seriously and have resigned as a MR reviewer, and

have sold my MR's. It is fine to suddenly feel the absence of pressure.

I was definitely getting stale and it was time to retire. Now I am

even making progress in my book, having finally got the heat equation

potential theory in a reasonable form. I cou&d have kept my office but

have no use for an office at the U except to hang my coat in winter, so

I am being transferred into an office for two. You were probably wrong

to vote for Jarvis-Gann because the needed money may be raised by

increasing the state income tax and that may hurt you more than the

property tax.

Algarve sounds interesting. I have never been in Portugal. We

are going off for a two week vacation to Canada fly to Winnipeg,

rent a car and drive around. We want a cottage around a lake full

of fish for me to catch and with paths for walking. In anticipation

I am buying a new rod and reel.

My book will be infinitely heavier than Rao's, which is very good,

and elegant, but short on general theory. But I am hoping to refine

my treatment to the point where it is reasonably understandable. I am

absolutely rigid on not going beyond heat equation and Laplace's

equation.

Bet you never heard of the following translation of a potential

theorem. If X and Y and Z are positive supermartingales with X there are positive supermartingales Y1 1Y, m Z1

/^1 1

a^-c--







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801


February 16, 1979

Dear Chung:

Who is this guy? Should I have heard of him?

I have reservations for me and Elsie on that plane from Zurich.

Like you, Elsie will probably come a week ahead to wander around Switzerland.

She has asked the national Planned Parenthood office to send her a letter

saying how good she is etc. and will send you a letter saying what she

wants to do in China with a copy of the PP letter enclosed. I assume that

credit cards are worthless in China but that travelers checks are accepted.

And I also assume that my Norelco will work on Chinese electricity or I shall

return to capitalism looking like Karl Marx. I have been reading about

China and am amazed at how often Mao had to struggle to stay on top. It

also seems to me that whenever his country was doing pretty well Mao got

another hare-brained scheme to disrupt it such as the Great Leap or

the Cultural Revolution. I expect that by the time we arrive the Chinese

will officially agree with me.

Regards,


-^







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801

March 5, 1979
Dear Chung:4 C 4 .e-t
I am surprised by the ms. you sent. I thought you were going to
assume lots less I proved that a superharmonic function is continuous on
Brownian paths as follows. Cartan had proved a Lusin theorem using capacity
instead of measure, and I proved that Brownian paths are not likely to hit
a
sets of small capacity. This means that Brownian paths remain in iba set
on which the restriction of the given superharmonic function is continuous
aside from paths of small probability. Of course a probabilist doing this
and wanting to avoid capacity ideas might prove Cartan's result using
hitting probability instead of capacity. I use a different method in my
book. I prove right continuity first: trivial if the superharmonic function
u is continuous; a superharmonic function is the limit of an increasing
sequence of superharmonic functions continuous on an arbitrary compact set
so I apply Meyer's theorem that a process which is the limit of an increasing
sequence of right continuous supermartingales is right continuous to get
right continuity of a superharmonic function on BM. Next I point out that
this is true no matter what the initial BM distribution is, even if the
initial distribution is Lebesgue measure on the whole space, in which case
the izftial distribution is the same as the distribution at any later time.
This means that if c > 0 and w(.) is the BM then w(c-t) is also BM for
t < c so u on the reverse BM is right continuous so u is continuous on
the BM. There are minor technical points to handle here but easy. For
instance since the function u is given only on an open set D I have to prove
(trivial of course) not only that u on BM is right continuous up to the
first hitting time by a B path of the boundary, coming from inside, but
that u is right continuous on the paths whenever the paths are in D.
I never considered any processes fancier than BM. It is amusing in
retrospect that Hunt in writing his original versionlnew so little of
martingales that his excessive functions were right continuous on paths but
he had nothing on left limits; I was Ill. J. editor and pointed out the
left cx=n y. I treat balayage in my book first straight potential
theoretically and then give the probability version. It seems to me that
both points of view are necessary for full understanding.
The liaison office sent me visa blanks but the instructions are to
send visa applications in a month before departure so I have not yet returned
them.







I am now doing h-paths in my book and as usual keep finding that
my papers are so obscure that new methods are essential. The U. of I.
Press wants to publish an assortment of my papers, unfortunately having
been encouraged in this idea by Burkholder and Meyer. I am dragging my
feet on the project as much as possible because I would refuse to allow
such a publication without correcting the mistakes and fixing the
'obscurities' and this would be a lot of work. When I see all the
nonsense I have published I am horrified at the prospects for my new book.
I have had to learn lots of things I never went through before, like why
the a algebra families of h paths are automatically right continuous etc.
No doubt half my treatments are inappropriate.

^Z-^-9-;







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801


March 8, 1979
Dear Chung:
You probably won't believe this but when I wrote you that of course
I had originally done nothing beyond Brownian motion I had forgotten all
about the heat equation and that I had used different methods to get
smoothness there than for Brownian motion' Hunt is right. His right
continuity of excessive functions on paths is proved just the way I
proved right continuity of subparabolic functions on space time BM paths.
It makes essential use of the first hitting time when the change is > s
and is based on proof of continuity at time 0.
Enclosed is my latest version and perhaps even the final version of
one of your favorite topics, bD is the Brownian transition density
function in the domain D (that is BM killed when it reaches the boundary)
h
and b is the h Brownian motion in D.
I heard yesterday from Kaufman Buhler that Peters and wife have quit
Springer and deduced from the way he talked that there must be unpleasant
quarrels involved, about which he did not enlighten me. Have you heard
anything?
Regards,


14 i-G







10. Capacitary distributions in terms of last hitting times* Let
D be a Greenian subset of E and let h Gd b be a potential on D. Let
w(.) be a Brownian motion [h Brownian motion] in D from a point with
probabilities and expectations denoted by P, E EPh, ?] and with lifetime
S. According to Theorem 4 the limit w(S-) exists Ph almost surely, is in
D, and has distribution GD( ,T)(d1))/h(F). In particular suppose that
h is the capaoitary potential of an analytic subset A of D. that is
(Theorem 7) h(() is the probability under P that A is hit &a a strictly
positive time by v(.). Then L is the capaoitary distribution, which is
thereby ovb*-&.d. iA ftems of the aaymptotio hitting distribution of
h Brownian motion. The oapacitary distribution can also be obtained
as an actual last hitting distribution of Brownian motion. In fact if
h is the apacitary potential of A the Brownian motion w(.) in D from '
if killed at the last hit of A, becomes a Markov process on the parameter
interval J]0, o with transition density b- according to Section V.15.
According to that section this killed process can be identified with
h BV0onian motion in D from F except that all Ph unconditional probabilities
are to be multiplied by h()). It follows that the distribution under P
of last hits of A by w(.), under the hypothesis that A is hit, is
GD(1'n)u(dT)/h( ), the same as the distribution of w(&S) under h found
above.
Roughly the following correspondence has now been established:
first hitting distributions by Brownian motion give harmonic measure;i
last hitting distributions give capacitary measures.








Application to continuity properties of sample functions. Let

{x(t),W(t),t > O} be a progressively measurable process on (a,Q,P) with
state space the extended reals and suppose that the restriction of P to
each a algebra '(t) is complete. Let x'(w,w), x"(.y,w) be respectively
the lower and upper envelopes of the sample function x(,w),
x'(t,w) = lim inf x(s,W), x"(t,w) = lim sup x(s,w).
6-->0 s-tl < 6-->0 js-ti : 6

If b > 0 let Tb-b be the entry time for the progressively measurable
process {x(b+'), (b-+)} of the set ]a, 0D] for some specified a, that is
Tb is the infimum of the times t _>b at which x(t) > a. Then Tb-b is
optional for 7 (b-t) so Tb is optional for f +(6) and

{(: sup x(s,o) < a} = ({Tt6 > t+6} (- +(t+6).
is-t < 8
It follows that x"(t) is f+(t) measurable and that the process (x"('), +(%)}
is progressively measurable. Similarly x'(t) is +(t) measurable and

{xt'(t?), ()} is progressively measurable. The sets
{(t,c): x'(t,w) = +oo}, {(t,)): x"(t,o) = +ao}
are therefore progressively measurable so if we define
x'(t,w) = x'(t,c) when x'(t,z) +oD
= otherwise
xo(t,w) = x"(t,w) when x"(t,w) oa
= +aD otherwise
the processes (xo( ), ()), {xo'(),,+(Q,)} are progressively measurable,
as is the difference process {x"I(.)-x'(f;),+ (T.)). If T is the entry time

by this process of.the set ]0, ao], T is optional and the set {T c} is
the subset of Ql yielding sample functions continuous and finite valued
on the interval [0,c[. This argument is also applicable with x'(t) and
replaced
x"(t) respectively by the limits inferior and superior on the right at t.








If the state space of this progressively measurable process is

Polish, metrized by a distance function d(3,,), paired with its Borel sets,

the process {d(x(0),x(t)),7(t),t > 0} is progressively measurable and for

a >0 the optional time inf{t: d(x(0),x(t)) _.a} is useful. An
adaptation of the preceding discussion of extended real valued processes

can be carried through as follows. By hypothesis the functions

(s,t,w)o->x(s,w), (s,t,w) o-->x(t,w) are measurable from the space

[0,c] >< [O,c] >X 2 paired with [O,c] x><[0,c] X< (c) into the
measurable state space. Hence the function (s,t,w)e -->d(x(s,w),x(t,w))

is a --4-- measurable function from the first space into the reals.

It follows that the set

I(s,t,w): d(x(s,w),x(t,w)) > a}
is in B[O,c] x><[0,c] >< (c) and (Appendix

that the projection of this set on l. is analytic over 7(c) and so is in

-(c) by Lusin's theorem. That is

{w: d(x(s,w),x(t,w)) >a for some s, t in [O,c]} (-& (c)
so sup d(x(s,w)xC(t,w)) is 7(c) measurable. Slightly more generally
s,t c
a trivial variation of the argument shows that the oscillation of a

sample function on an interval [b,c] is -(c) measurable and it follows that

if y(t,w) is the oscillation of the sample function x(Q,w) at t the process

{y(t),' +(t),t > 0} is progressively measurable. If T is the entry time
of this process into the interval ]0, c[ this time is optional and {T > c}

is the subset of L. yielding sample functions continuous in the interval

[0,c[. As in the case of an extended real valued state space this argument
is also applicable to one sided oscillations, so that one can derive the

measurability of the subsets of .L corresponding to the sets of sample








functions which are continuous on an interval or are right continuous there
o0f e"Cceedi'Mj
or have right limits there or have oscillation 'm Cset some assigned value

there,... Observe however that in this discussion there has been no

assurance that another progressively measurable process with the same

finite dimensional distributions will assign the same probabilities

to the subsets of its probability space discussed above as the process

under discussion. For example there has been no assurance that the finite

dimensional distributions of x'(c), x"(G ), y(.~) or that the distrihtions

of Tb, T will be independent of the choice of (n, ,P). In fact examples

in Section 11 show that such an assurance would be false. If however

{x(t),t > O} is given as a right continuous process and if the family

Y(.) is either already assigned and has the desired properties or if

;(t) is defined as the a algebra generated by the null sets and Y{x(s),s < t}

then all of the work in this section is applicable in this stricter context

and the probabilities mentioned above will be the same for any other choice

of the original process x((D) with the given finite dimensional distributions

as long as the process is right continuous (or even almost surely right

continuous),







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA. ILLINOIS 61801

April 23, 1979
Dear Chung:
Thanks for the corrections. Although what I sent is just from my first
slightly smooth rough draft I might well have missed those bloopers permanently.
But if I replace s by c/2 the continuity proof is still OK. I like my
proof because it shows what the issues are without any clutter, besides
proving the theorem. I admit of course that there is a simpler more
sophisticated proof using the section theorem of the more general result that
for a progressively measurable process there is right continuity if T 4 T
implies that X(T )-> x(T) for T optional. But so far I have had no need
for section theorems and so did not want to use that proof. I have tried to
make the book selfcontained and have all the necessary analytic set stuff
in an appendix. One of my ideas was to make Strasbourg reasonable by
showing how that point of view was not only necessary for a proper treatment
of Brownian motion but that it is not so hard. I enclose a couple of pages
to show how easy it is to get the optional times that worry you. Incidentally
I can use a more messy proof and avoid analytic sets in proving Meyer's
theorem. Perhaps I'll include that for pedagogical reasons.
Of course you are right that it would be a fine idea to have a faithful
reader go over everything. But I know no such victim'
You should enjoy some of what I am sending since as you see I make a
point out of the problem of finding conditions under which trick distributions
for one version of a process are the same for another version.
Frank Knight wonders why you never acknowledged a proof he sent you
of something or other involving the Poisson Summation Formula.



^7--,,i













August 7, 1979
Dear Doob,
Gerald Goodman was here and told me your instructional lectures
fell on dead ears. They wanted to know what measure space you were
in! I hope the Chinese audience was more slavish and got something
out of your lectures there. Your book may be a good introduction to
people who can read Smdlett, but I don't think this is the way most
people latch on to research. My stuff was of course aiso over- their
heads; but if the more capable ones can ,close the not-so-wide gap,
'they are already onto things they can lay their hands on.
What did you think of Goodman's talk?
If you think you konw all tlha is to know about Fatou bourndary
etc., try the following concrete little problem. .Let q be'bounded
measuable,(or as smooth as you want), and put
u(x) = Ex(exp TQ(tf l)dt)
where T is the exit time from a bounded open set Gi and the function
u defined In G can be shown to be a solution of the: Sch.rodinger equa-
tion provided it is finite at one poiAt in G. Take. G: to be a ball even.
Now you think (after finishing your tome) you know',e4erything about
this old bore. OK, can you prove that when x ap 'roaches the boundary
(a s here) from inside G, the limit i li1 come i1 Taut? POf course,
along almost every path this is indeed so. Does your Fatou stuff help
any ?
I assume Dr Field got my second letter and would send the copy of
the photo I asked. Any news about the Chinese whpeb:'re coming to Urbana


Sincerely,







P The result becomes trivial if v(x ) oo where v, b like u but
with T replaced by the exit time from a larger s $, \e is is like
the dirichlet problem, but the corresponding DPf oR' itself says
nothing here.






STANFORD UNIVERSITY
STANFORD, CALIFORNIA 94303
DEPARTMENT OF MATHEMATICS


September 24, 19,79.
Dear Doob,
How are you doing with my problem? I can now do a lot more, and
it is surprising that nobody seems to have thought of joing such things.
With some mild condition on 6G, for example if it is 0 I can show
that if e
x( exp q(X(t))dt)
is finite at one point of Ga. then there is an open neighborhood H of
4 such that if the T(G) above is replaced by T(H), the resulting func-
tion of x is bounded in H. Keep going a transfinite number of times to-
obtain a maximal domain for which the corresponding function is now
infinite everywhere, or e.Be T(G).may be replaced by oq and the corres-
ponding function is finite everywhere. I- don't know if the latter can
happen .. This should yield a theory of-extension of positive solutions
of the Schrodinger equation and must have to do with the classical .igen-
value problems, probablyy the variational theory. Unfortunately I have
nearly forgotten that I ever knew about such:things and am rereading parts
of Courant-Hilbert to see what those old fogies did. It is a physical
as well as mental pain for me to read that sloppy book but I don't know
any other readable source which gets to the heart of the matter quickly.
Did you give people that list of publications,containg some 80 items?
If so )you better revise some of the mistakes, e. g. the co-author is left
out in, N8. 66 (I think). Some of your choice seems foolish too. It may
amuse Vou that one of your fans still cannot think of.anything el e you
ever d4d except your 1.936 paper using an outer probability one to do the
continuous papameter processesr--later revised by Andersen-Jessen and
Tulcea! Is the history right there? I know there was a big mistake noted
by Dieudonne and a special case was spelled out by Tdlcea, quite trivial.
But what .about that A-J. business. D-.d they really correct your funda-
mental theory too? That guy seems to think so.

Sincerely,












COPY
















January 10, 1980


Professor Kai Lai Chung, Mathematics Dept. Stanford University
Professor Charles M. Stein Statistics Dept. Stanford, Ca 94305



Dear Charles, Dear Kai-Lai,

The following is the text of a brief letter to Doob congratulating
him on his National Medal of Science.

This is to congratulate you on the award of the
National Medal of Science. We strongly feel that
this award makes an appropriate recognition of your
profound influence on probability theory that you
have been exercising over the last forty years.

Congratulations and best wishes,




Jointly with David Blackwell I plan to have this text drawn on top
of a piece of paper, perhaps 2x3 feet in size, and to collect enough
signatures to cover the space below. The sheet will be available
in our office (Rm. 401) for some two days, perhaps Monday and Tuesday
of the coming week. Next, the sheet will be put in a nice frame and
sent to Joe. He may like to have it hanging on a wall of one of his rooms.

Because both of you, Charles and Kai-lai took an active part in
nominating Joe, it would be natural for you to wish to sign the
congratulatory sheet. When the time is ready for signatures I will
inform you. However, there is also the possibility that the people
in the Math. and in the Stat. Departments at Stanford will wish to








=2-


embark on an action similar to ours. I am not sure who might be the
individuals interested in some such action. If you have some thought
in the matter, do not hesitate to inform the relevant individuals
of our plans.

With best regards,

Cordially,



Jerzy Neyman







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801

February 2, 1980

Dear Chung:

I am just reading Chung-Rao, and assume and hope what you sent is not

a reproduction of what you sent Brelot. Incidentally whoever typed that should

clean his typewriter keys a trivial job.

p. 1 The definition of e(t) omits exp

Theorem 1. If D need not be bounded then ao may be a boundary point

and this fact will come in repeatedly. For example is uf bounded (defined ?)

there? Is CD a regular boundary point? (Always in dimensionality > 2.)

Theorem 2. Is z allowed to be ao ? I don't like the way you phrased
by probability
the reference to me there. The solution of the Dirichlet problem goes way

back. See for instance Courant-Friedrichs-Levy in 1928. Kakutani discussed

it for a smooth boundary. In fact it was Kakutani's work that made me look

at the problem. I was the first to do it for nonsmooth boundaries and I

got the condition for boundary point regularity.

p. 5 I don't understand what you are driving at in the sentence

starting line 4: L'extension ... Didn't' Kac have bounded domains and therefore

have optional times?

Theorem 4. If by sousdomaine you mean an open set, it should be

RELATIVELY compact in D.

p. 3 Corollary 1. 2x'=p xre tx'tf You should realize in writing here

and elsewhere that when nonprobabilists have a domain D they think of regular

points of the boundary for the Dirichlet problem, not regular points for

the complement. It is unfortunate that there is this difference in language.

p. 4 I'll bet you want D bounded. You should make clear whether OD
n
is counted in closures. And OUCH !: I see that you do not consider probability

as a part of analysis. Welcome to algebra' Or are you a geometer? A topologist?








I had thought of your proof of Lemma 2 but somehow thought there was

some difficulty there, I forget what by now.

On rereading your letter I see that the typing is yours. Buy some

STAR type cleaner, putty like stuff that one just pushes on the type keys

and then pulls off to get the ink off the keys. 2

Bob Kaufman's wife (Chinese) tells me that Hu told her that the

mathematicians here were not interested in his kind of probability. I shall

try to get him to give a seminar so we can tell what he really does. He

writes on M. Chains but of course I can't tell what he has, if anything.

I have no objection to your publishing that discrete proof relating

superharmonic functions with supermartingales. As far as I know there are now

three proofs of that: by Ito integrals; my first proof done by extending the

functions to the whole space; the discrete proof. The latter proof is cuter

but Ito integral proof is cleaner in some ways.

Neyman asked me to send him a Xerox of the title page of the Russian

translation of my SP book. Apparently he is going to write in favor of not

being nasty to the Russians and that there is and has always been international

collaboration between scientists. The title page is somehow supposed to be

evidence; why he needs that for proof is beyond me. Lorch was around Stanford

recently. Is he organizing another letter in favor of detente and being nice

to Russians? It is amusing that noone seems to recognize what I missed also

at first that Dynkin and colleagues are not trying to attack the Soviet

regime but merely a small number of individual mathematicians in positions

of power'













April 6, 1980
Dear Doob,

Try to prove that B1 stopped at the boundary of a bounded domain
is a Hunt process. The state space is the closure and there may be
irregular points. You need Kellogg's theorem! or else you made a mis-
take. For killed process it is easier, but contrary to what Mayer
did in his lecture notes, where the 6 is any old points it MUST be
the one point infinity (Alexandroff) or esle wrong again.

I 4 came back from Gainsville where there seemed to. be some
real interest from young guys in my Feynman stuff. The physicists
had ever thought of tbis even though (you know) that the Dirichelt
problem caNe from eletromagnetism. That shows how behind they are.
note in- C. R. should appear soon.

Do you ha''e this in your tome? For BM in the 'lane I have a nice
.approach using the a-potential density but need a old goo expansion




It is important that o(l) is uniform for x in a compact when a tends
to zero. Tv,'is rust be well-known to Bessel function experts so I do
not try to prove it directly. But maybe you have used it. I will
read. your reYis ~d stuff after I get my breath back. My version of
your ptiff ( Hunt) will soon be typed and I'll send a copy to you.
But frankly I do not want to make any unnecessary revision from here
on. It is a credit to you (medal or no) that you are still.doing some
nitty-gritty mathematics, whereas your brethren such as Kao and Brelot
are probably wniing and dining their declining days. I almost said that
inn.y~y talks in Gainsville. The young 'uns approciAte that. I am encour-
aging them to hire Rao who wants to leave Denmark and they seem very
eager to do so ,NSF objected to having foreigners on the sign on the wall. If they ask you I hope you writ% a i4S-than-
usually-faint letter S4efSRao. Despite his faults he is probably as
good a probablist as the next one around.


Sincerely.












April 8, 1980


Dear Doob,

I doped out the probability solution of Poisson's equation
in the planes namely

Ex(PT f(Xt)dt) satisfies the equation
if f is Holder continuous in D, etc. Such a proof is not given
anywhere I have looked, Port-Stone or Rao, but I wonder if it is
in your old papers? I use exactly the same method as in space but
must start with a-potentials, use the asymptotic expansion (found
in tables H~xtx gra ). let a go to Os examine a number of error
terms, and the result emerges. Not hard but rather messy. But this
shows that that the logarithmic case is in principle no different from
the Newtonian, also the necessity of the Laptace tranform. If it is
new I'll publish it in an applied journal. It turned out too long
for my book.

The tricky approach used by P-S. does not identify the solution
to be the Expectation above, nor does Rao. For 1 dimension I have
an extremely elegant (NEW) proof using just the Schwarz generalized
second derivative.

Sincerely.



I am curious whether your tome will contain a detailed proof
that if f is Holder continuous and has compact support then Uf is C
This is given in Kelloggs a sl. more general version in P-S and Rao.
I decided that it did not belong in my book, but will have to cite it.







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801

May 22, 1980

Dear Chung:

My complaint about your ms. is not what you do or do not do but that

you give a false impression to readers. Everyone who knows anything about

differential equations has a standard definition of what "regular boundary

point" for the Dirichlet problem means. If you are going to use regular as

some probabilists do you should point out that your definition is in fact

equivalent to the d.e. definition but that you are not going to prove this

equivalence. Of course in your definition boundedness of B is irrelevant; in

the d.e. definition boundedness creates an extra difficulty.

I don't understand your question about semipolar = polar. Brelot proved

this by proving that a set is polar if no point of it is a fine limit point.

Such a statement would have been meaningless to Kellogg because the fine

topology was after his time. Kellogg in his book refers to himself as proving

that every compact boundary nonpolar set has at least one regular point. From

a modern point of view this implies that the set of irregular points is polar.

Of course Kellogg was referring to regular in the d.e. sense, equivalently

his points are regular if there are barriers, as he proves. I should have said

that he refers to himself in the two dimensional case and says the result is

not known in other dimensionalities.

I .don't understand your remarks about the Poisson formula. What I do

in my book is the classical business. Using inversions one can write down the

Green function of a ball and this gives the Poisson formula. Don't you like

inversions?

When you write "maximum principle do you mean what I call "domination pr.? "









The terminology is confused in this area. The theorem I mean is that

a Green potential majorized on the support of its measure by a positive superharmonic

function and finite on the support is majorized everywhere by the superharmonic

function. By Evans-Vasilesco one can assume that the potential is continuous

and that its measure has compact support. Then one can use the fact that on

the complement of the support of its measure the potential is continuous,

harmonic, and has limit 0 at the boundary and limit
majorant on the support of the measure. That settles it. If this is what you

are after I can send you the proof, a little clearer than my confused description'

I enclose my proof that -the composition of a superharmonic function

with BM gives continuous sample functions. I don't mind your not proving this

in your ms, but it is unfair to the reader to get right continuity and not

mention that more is true.

I sent a strong recommendation of Rao to Gainesville.



>^-Z>-^ -^






June 6, 1980


Dear Chung:

You have not rewritten your stuff often enough!!!' On your 4.85c

you should of course have W independent of Y. And where does W come from?

You have to make a product space and then use Fubini etc. Unless I just had

this inspiration: you are looking for a continuation of X(c-t), that is some

increments to add on to it. Why not use the dX for t > c? You do not need

them for anything else.

Every time I type a page of my book now I date the page, and next time

copy the old date along with the new one. Lots of my pages now have 5 or more

dates! I can't remember when I started on this damn book but my oldest dates

(I only started my dating system after some time of work) are in 1976.

I don't know how to answer your question about Prop. 11 since I don't

know how you defined 'solution to the generalized Dirichlet problem'. The

general result is: The Perron etc method leads to a unique result ( "solution")

if and only if the boundary function is in a certain L1 class that for

harmonic measure. Probabilistically this is the case in which your w(t) in

(29) is a uniformly integrable martingale. There is a corresponding result

proved in a more complete book which will be published in the year oo that for

a boundary defined by an arbitrary compactification the Perron method can be

carried though as usual but harmonic measure is on a sub algebra of the Borel

boundary sets, and Brownian paths do not necessarily converge to single points

of the boundary, but everything above goes through. For the Martin boundary

a harmonic function is a Dirichlet solution in this sense if and only if it

is quasi bounded that is if and only if it is the difference between two positive

functions each'of which is a countable sum of bounded positive harmonic functions.

Incidentally I'll bet a nickel or so that you have not justified jumping

around between initial distributions.

I shall be east vacationing with Elsie June 7-17.




>$KZ^4






June 6, 1980


Dear Chung:

You have not rewritten your stuff often enough!!!! On your 4.85c

you should of course have W independent of Y. And where does W come from?

You have to make a product space and then use Fubini etc. Unless I just had

this inspiration: you are looking for a continuation of X(c-t), that is some

increments to add on to it. Why not use the dX for t > c? You do not need

them for anything else.

Every time I type a page of my book now I date the page, and next time

copy the old date along with the new one. Lots of my pages now have 5 or more

dates! I can't remember when I started on this damn book but my oldest dates

(I only started my dating system after some time of work) are in 1976.

I don't know how to answer your question about Prop. 11 since I don't

know how you defined 'solution to the generalized Dirichlet problem'. The

general result is: The Perron etc method leads to a unique result ("solution")

if and only if the boundary function is in a certain L1 class that for

harmonic measure. Probabilistically this is the case in which your w(t) in

(29) is a uniformly integrable martingale. There is a corresponding result

proved in a more complete book which will be published in the year ao that for

a boundary defined by an arbitrary compactification the Perron method can be

carried though as usual but harmonic measure is on a sub algebra of the Borel

boundary sets, and Brownian paths do not necessarily converge to single points

of the boundary, but everything above goes through. For the Martin boundary

a harmonic function is a Dirichlet solution in this sense if and only if it

is quasi bounded that is if and only if it is the difference between two positive

functions each of which is a countable sum of bounded positive harmonic functions.

Incidentally I'll bet a nickel or so that you have not justified jumping

around between initial distributions.

I shall be east vacationing with Elsie June 7-17.


/^--d^"^ Z^^7














August 1, 1980
Dear Doob,

I intend to "close" my bod& Volume 1, soon so these may be
near-final sections except possibly a section on my latest Feynman-
Kac stuff by popular demand. ihese sections contain much is new
and developable. The idea of a quitting kernel goes a long way and
is THE correct way to treat these matters of classical fame. To con-
ceal the i:eneral structure by some silly special case (such as your
and Rao's proof of ,(14 on p. 115) runs directly contrary to the modern
spirit. It is tantmUn tn t proving your supermartinagle stopping the-
orem in some stupid independatcase (as Robbins et al actually did some
time ago, not to revert to Wald). These are strong words but there is
some truth to it for anyone who has the humility to learn N "new trick.
Actually Agemra did a lot of deep things on the san.e .ereral theme
but he (being of the !.rench school) went at once to additive functional,
without coring down to measures of potentials which are all I knew. The]
again, a mix of old and new is more to ny taste---and also In the grand
tradition of i;r thematical development!
Anyway, let me see if you have any "usable" comments on these be-
fore they are corrected. My only trouble now is having made such a
wide swing I don't know how to get back to BM without a long trudge.
[Miusicians say this of the ending of Beethoven's symphonies---there is
some truth to that too.. Perhaps I will just add a couple of pages
and relegate it to an old story to be found in books such as yours.
I certainly don't have the patience to go through all the c asical
stuff as particularly applied to BM. But there is some new stuff which
will yet be added to the last section.


Sincerely,






UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA. ILLINOIS 61801

August 7, 1980

Dear Chung:

Has it ever occurred to you to join MEGALOMANIACS ANONYMOUS?

Of course I do not know what you have covered in the rest of your

book but I have a feeling that you are not tying things together very well

for students unfamiliar with potential theory. For example on p. 4.116

I assume it is obvious that the left side of (20) defines an excessive function

when B is fixed. Thus (20) is the usual representation of a potential whose

2 measure is supported by B. You might point out why you can't just say this,

why the measure is supported by B,.... Or am I missing something?

On p. 4.117 you mention Robin's problem. The place I know of where

Robin's problem is mentioned most, with lots of references to his name, is

Nevanlinna's Analytic Functions (Springer). He goes into lots of detail etc.

but the kernel he uses is the logarithmic kernel everything is in two

dimensions and of course this is not positive and does not come under your

work. Perhaps you should warn readers about this. Incidentally here is

the log case as I shall do it in my book: Suppose that D is an open plane set

which is a neighborhood of the point 0, deleted neighborhood I mean, and let

G D( D,.) be the Green function of D for the pole at cD. This function is

harmonic on D and is like loglzJ near D; it exists if the complement A of

D is not polar, and I assume this from now on. It is easy to extend GD( C,.)

by 0 on the interior and quasi everywhere on the boundary of A to get an

extended function defined on the plane and positive and subharmonic there.

The associated negative measure -p has the property that its logarithmic

potential solves the Robin problem. More specifically: if G(g,x) = -loglg-,l

then Gp is a potential with constant value c quasi everywhere on A; c is

Robin's constant.







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801

August 7, 1980

Dear Chung:

Has it ever occurred to you to join MEGALOMANIACS ANONYMOUS?

Of course I do not know what you have covered in the rest of your

book but I have a feeling that you are not tying things together very well

for students unfamiliar with potential theory. For example on p. 4.116

I assume it is obvious that the left side of (20) defines an excessive function

when B is fixed. Thus (20) is the usual representation of a potential whose

Measure is supported by B. You might point out why you can't just say this,

why the measure is supported by B,.... Or am I missing something?

On p. 4.117 you mention Robin's problem. The place I know of where

Robin's problem is mentioned most, with lots of references to his name, is

Nevanlinna's Analytic Functions (Springer). He goes into lots of detail etc.

but the kernel he uses is the logarithmic kernel everything is in two

dimensions and of course this is not positive and does not come under your

work. Perhaps you should warn readers about this. Incidentally here is

the log case as I shall do it in my book: Suppose that D is an open plane set

which is a neighborhood of the point oD, deleted neighborhood I mean, and let

GD( D,.) be the Green function of D for the pole at CD. This function is

harmonic on D and is like loglzl near a); it exists if the complement A of

D is not polar, and I assume this from now on. It is easy to extend GD( c,.)

by 0 on the interior and quasi everywhere on the boundary of A to get an

extended function defined on the plane and positive and subharmonic there.

The associated negative measure -p has the property that its logarithmic

potential solves the Robin problem. More specifically: if G(,T|) = -loglC-1l

then Gp is a potential with constant value c quasi everywhere on A; c is

Robin' s constant.








And why not say somewhere under what conditions one talked about harmonic

functions and has a Riesz decomposition of any excessive function into the

sum of a potential and a harmonic function; and under what conditions an

excessive function majorized by a potential is a potential,.., and so on.

Or maybe you have all this someithere.

On p. 4.130 why does K have to be compact? Aren't your measures

inner regular? And isn't you space nice enough so that all your measures

are finite on compact sets and therefore a finite?

Incidentally I don't like your name "quitting time ". If B is a ball

and a BM is started from a point of B the natural meaning of quitting time

would be the exit time from B, not the last exit time. People say that they

are quitting a place FOR GOOD if they never intend to come back.

I am disturbed that I do not seem to find the word excessive in your

text so that the potential kernel is a bit mysterious. Have I missed some

extra generality that you are assuming?

As ever (no megalomaniac I)






Do you know anything about this? I had a cable from Ke Chan Kuo

who professes at Southwestern Jiaotong U., Enei, Sichuan. He and a couple of

colleagues will visit Urbana in September. All I know about him is that he

took a couple of courses from me in 1947 and left with an MA.


9jI V K






UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801

October 6, 1980

Dear Chung:

What makes you think that anyone ever takes your advice? NSF or

whatever paid no attention to you and never asked me about a Princeton Institute.

Actually I am not sure what you are writing about. I assume that it is about

that Institute that Cambridge etc. are fighting to get. It would be silly

to ask me since it is well known that I am on the Board of Trustees of the

Institute for Advanced Study so cannot be expected to give an unbiased opinion

I am against such an institute on the principle that no matter what is said

the money will come out of the money that is now paid for the usual contracts.

A fundamental (easy) result that makes life easy in the classical context

is that an h-path Brownian motion in an open set D can be constructed as follows

if h is a potential, h = GDX. Let w(.) be the h-Brownian motion, with lifetime

S. Then if the intiati point is 9 the left limit w(S-) exists almost surely

and the distribution of this left limit is GD(,q)X(d')/h(). In particular

if h = GD(.,1) the paths go from to %. The general h-path process for

h = GDX can be constructed as follows: pick the* endpoint T using the above

distribution and then pick the path to T from g using the special case h = GD(.,TL)

In other words the conditional distribution of h-paths from %, given that

w(S-) = I is the distribution of GD(.,T)-paths. This fact makes lots of things

easy, including the existence of left limits at the path lifetime because this

is easy for GD(.,)) paths. Is there nothing in your context that looks'like

this?

It is a fine idea to leave stuff dangling, but not in a book written

for students. I never realized fully before how hard it is to organize hitting

and so on in a logical reasonable way. I was horrified a few days ago to find

that I have been working seriously on my book since early 1974.



( \-s---i-







Oct. 23, 1980
Dear Doob,
I am writing a few bibliographical notes for my book. As usual
with authors, I cite myself liberally and credit the others only
by their books, of which ybu have one. [Meyer has many.] However
I want to add a few genuinely historical comments. I found your
proof (H !) of the right continuity of subparabolic on brownian paths
in your paper on Heat. To begin the argument you-had an upper semi-
continuous function u there but Hunt had to use a bit of fine topolo-
gyfor a general eneessive function. That is not quite trivial is it?
Now I can never locate where you proved its continuity (at least for
superharmonic, since I don't know about subparabolic). This is call-
ed Doob's theorem but where is? The proof by reversing the time (we
discussed this recently) is given by Meyer in his L. N. Intuitive as
it seems, it is quite tricky conceptually as I hope I convinced you
last time. Did you or someone else use this argument? [I recall you
proof of continuity used some sort of capacity, so not by reversing.]
Since the continuity yields the Kellogg-Evans theorem was it noted
by the Brelot school? You know at thatime Kellogg wrote his book he
had the theorem only in the plane.
My reaJldiscovery is that you did not solve the Dirichlet problem
as such. W was admitted by you on p.1 of your Semimartingales
1954 paper. Then who proved that u converges to f at a regular bound-
ary point where
u(x) = EX f(Xp)) f continuous, T= exit time
from (bounded) domain
To amwhItIa*.a bit farther, nowhere in your papers you ever proved
that Yhe u above is harmonic. [The Gauss averaging theorem is zNrer
mentioned.] On p. 112, loc. cit., youjssumed that u is the solution
of the Dirichlet problem (a la Brelot) and then showed a stochastic
convergence of u(X ). All this must be awfully confusing to the un-
initiated. On hindsight I suspect that you were at that time so
absorbed in the Brelot stuff that you missed the very simple probabi-
lity way. See p. 97, last few lines. Was it Dynkin who first wrote
down the solution? In my book it is now just a page or so.
The question for me is: since it is so easy to get the whole
thing from probability, what were you doing in those papers citing
all that resolutivity etc.? What is gained? At least for the New-
tonian case, I can see nothing gained in the generalities you dealt
with, later again in the Berkeley paper. If you look at either Rao
or Port-Stone you can see that the probability method does everything
neatly and there is no need to drag in the nonprobabilistic stuff.
The same thing has now been done for Schrodinger. The results
contain all the results from PDE. Here is a little teaser. Did you
know that if the domain is snall enough there is always a unique solu-
tion, (note: no maximum principle for Schrodinger)? Try to prove it.
But the probability solution is trivial becAtse if
EX(exptQT))< o where Q is a bd for,. below,
then
u(x) = EX{exp/Tq(X(t))dt.f(XT)}
is bounded by domination ana therefore is a solution with boundary
value f. See Theorem 4 in our p e' but the calculation is easy
and the harder Theorems (1 and 2 aF e not needin this case by domi-
nation. Now pup osea foluti-on is;given-n any-Bound-d domai Make
tka'tte'r e nio of fi te umber f -mal-one 4~-e -6







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA. ILLINOIS 61801


October 27, 1980

Dear Chung:

The essential difference between us in our attitude toward the books

we are writing is that you are a pure probabilist and I am not. I started out

in analytic functions and consider myself a probabilist-potential theorist.

The fact that a few potential theoretic results come out easily after a

certain probabilistic background has been developed does not make the potential

theory either trivial or uninteresting.

Subparabolic functions are upper semicontinuous by definition so I

could get my right continuity of the composition of a subparabolic function

with space time Brownian motion slightly more easily than Hunt.

I proved that superharmonic functions composed with BM give continuous

sample functions in my TAMS 1954 paper (Semimartingales etc.) by the following

argument. By a Lusin type theorem (Cartan) if u is superharmonic there is

a closed set such that the restriction to this set of u is continuous and the

complement of this set is a set of small capacity so BM does not visit it

much. It is a very simple argument.

I thought that I had convinced you that the time reversal argument I

now use to prove Mhe above continuity is not tricky!

Your discovery that I did not solve the Dirichlet problem is not one of

your better discoveries. What I proved was that the usual PWB method leads to

the probabilistic evaluation of the PWB solution. This means that for continuous

boundary functions the probabilistic solution has the right limit at a regular

point because the PWB solution has. Finally I proved the criterion for

regularity (that a BM from the boundary point hits the complement of the

domain right away). The funny thing is that I thought Kac had proved that and

I gave a reference to Kac in a footnote! You write that I did not solve the

Dirichlet problem 'as such'. If you mean that I did not simply write down the







the solution probabilistically and prove that it has the desired properties

you are right. But I wanted to do.much more. The PWB solution corresponds
Lebesgue integrable
to the Poisson integral of the given boundary function when the region is a

ball, and it is this generality I wanted, and which is given by the PWB method.

Incidentally the relative boundary is of course resolutive, that is continuous

boundary functions have the usual solutions. This is false if one considers

not harmonic functions but h-harmonic functions, that is (harmonic function)/h.

In this case the PWB method and the probabilistic method are still applicable

as I showed in later papers but it is not true that all continuous functions

are resolutive or that most boundary points are regular in some reasonable

sense, except for very special boundaries, e.g. the me=,ative boundary if the

domain is a ball, or the Martin boundary in the general case.

You should realize that everyone knew the probabilistic solution, even

for the heat equation, and even that the iterated log for BM is related to

boundary point regularity of the heat equation. It was probably Kakutani who

first stated the Dirichlet solution, at least for Jordan regions, explicitly.

You say that the probability method makes it unnecessary to drag in the

nonprobabilistic stuff. It could also be said that the potential theory method

does it equally neatly so why drag in all that probability stuff, which requires

much more space, since one should count the space devoted to martingales etc.

In fact some aspects of the theory are much more natural probabilistically

whereas others are more natural without probability.




/ \^.y 2












Nov. 11, 1980
Dear Doob,
If God had granted me better sight I'd certainly have dug into
the old literature more thoroughly for the fun of it. Anyway it is
now clear to me that Kakutani in 1944 proposed the (modest) prob.
solution of the D. P. without proving the convergence at a regular
point of the bdy. First he did not define regularity. Second he
stated the result for a Jordan curve which probably is false as this
is only continuous without any smoothness. It is probably for this
reason that he never published his proof. Unless he can prove the
result stated there he did not solve the DP. I have written him to
see if he has any other story. So far as your 1954 and later papers
are concerned it is also abundantly clear that you knew too much PWB
stuff and were chasing wild avant-garde stuff. It was well knowndur-
ing that epoch that many probabilists were sneering at your stochast-
ic boundary as they thought you proved only convergence along paths
(particularly in the Fatou theorem) and not along a given curve. You
might be well ahead of time but there was no inkling in your papers
of the modest solution of the DP as nowadays given in textbooks. That
should partly explain the confusion and resistance of other mathemati-
cinas to look at the method. No body doubted that you could have writ-
ten down the proof but I do not agree with you atht "everyone knew
the probabilistic solution". I happen to be around those days. Who
knows there may be some undiscovered papers out in the sticks that no-
one ever read.
The method I used to get a Riesz representation of Pyl is easily
generalized to excessive functions, as shown on inclosure. The idea
is certainly yours but the execution looks different because I assume
certain conditions on u to get the Riesz measure explicitly as shown.
Once more BM is too special to see what's happening. I am now wonder-
ing hwo to insert a little bit of this into my book MS. It would be
a nuisance to have to prove SMP etc. for the s-procaes even if all
is trivial.






STANFORD UNIVERSITY
STANFORD, CALIFORNIA 94305
DEPARTMENT OF MATHEMATICS
2/23

Dear Doob,
There is nothing new in Hsu's note but you seem to have missed
the point. The question is how to relate the log potential to BM
4aon J4L
in the plane as the Newtonian notentll to BM in bece. You betrayed
your misunderstanding b t asking why he did not do it in the latter caP
For the latter the O=potential does everything, but I HAVE TOT seen
anywhere a derivation of the killed potential in a ball by using the
approximation of a Bessel *-potential as Hsu carried out. If I had
seen this I would have used this method muchtmore transparent thar say
the one used in Port-Stone) to introduce the log potential. It is NOt
merely a question of getting the Green's function for the ball. But he
had to do it that way. If you know any other way (as will be given in the
catch-all book) let me see.
Hua tdld me that you mentioned a talk by the woman ian (?) on
the stuff of which Hou gave us a preview in Peking. Hou's is trivial
but they have written a whole book on it claiming applications to many
fields (Progpgine et al666ever heard?) Can you Five me an evaluation n
of her talk since I do not trust Hua's report. She is trying hard to
wangle some invitation but it would be embarassinp if her report iss
of the same calibre as Hou's.
Sincerely,








COPY







UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
DEPARTMENT OF MATHEMATICS
URBANA, ILLINOIS 61801
November 17, 1980

Dear Chung:

If you were not so proud of your ignorance of potential theory and

not so sure that everything can be written down explicitly in probabilistic

language without knowing anything about the potential theory context you

would not write me such nonsense. OF COURSE every boundary point of a

Jordan region is regular. A trivial way to see this is to note that the

region can be mapped conformally on a disk. There is also a standard criterion

for boundary point regularity in two dimensions: it is sufficient if the

boundary point is the endpoint of a Jordan arc not in the given domain.

And what is avant garde about the PWB method of solving the Dirichlet problem?

It is just the classical approach everyone knew sine about 1923, due more or

less to Perron. Wiener would have had it completely but he got a silly

counterexamplee" by a miscalculation and it was Brelot who first devised a

complete formulation. For a ball the method gives as solution the Poisson

integral of the boundary function and the class of PWB resolutive boundary

functions is the class of Lebesgue integrable ones, so everything is natural

and unavantgardish. The exact probabilistic counterpart is the discussion

of uniformly integrable martingales. I don't know what you refer to as the

modest solution of the DP. Of course the probabilistic formula for the PWB

solution is just the expected value of the boundary function on first hitting

place of Brownian motion. And of course since it is false that almost all

(harmonic measure) boundary points are regular when the boundary is for example

the Martin boundary, or when the boundary is the relative one but h-harmonic

functions are considered (harmonic functions/given positive harmonic function h)

no solution relying simply on regular points is sufficient to cover any reasonable

requirements. Thuse h-harmonic functions are solutions of very simple elliptic







PDE. And why should I care if others sneered at my solutions? I thought

more of my work then than others did and that was enough for me. Incidentally

the dual situation is true now: I seem to be slightly a patron saint of the

subject in some circles but I certainly think less of my own work than others

do now. Especially when I go over the chapters I wrote some time ago and try

to understand them.

What makes my book so long is that I want to make it closed: I want

it to contain everything needed. So for example I prove the SMP theorem

in a version strong enough to cover h-1rownian motion, and so on. Each time

I come to something now I have to go back to see that the preparation for it

exists in earlier work, and it never: does so I have to redo earlier chapters....



41
6 L/ ~~ P----







2/9/1992
Doob, The usual way is to postpone making a decision (why kept it?) in
the hope one does not need to make it eventually. Give the stuff to
Frank and ask him to store it away. I told him to ask you about your
enlargement of the natural tribes. I do not believe it can be done so
simply by adjoining "null sets" (p. 544, where defined???). This is a
serious misconception indulged in by hand-wavers. Compare it with what
people like Meyer doso see my "augmented" tribes and the exercises there
which should say you can't just adjoin some such N. Dynkin made the mis-
take too. Check Hunt's old paper. I would not have bothered to tell you
this trap but for the fact you are writing another book on measures.












101 West Windsor Road, #1104
Urbana, II 61801-6697


January 26, 1992
Dear Chung:
I have moved to a retirement home. Note my new address. The
telephone number is the same, 217-367-7029. As you can see from this letter I
have taken my computer along.
In going through my files I found our complete correspondence, over
two inches worth, all your letters and carbons of mine. I was about to throwla
the file away when I thought you might be more sentimental than I. Unless I
hear from you in a reasonable time saying that you want it, Ill throw it away.


,S6"----








101 West Windsor Road #1104
Urbana, Illinois 61801
March 4, 1992
Dear Chung:
I kept our correspondence, as I kept all my correspondence, on
vague general principles that some day I might want to refer to it.
Since neither of us has any use for it I'll throw it away when I
summon enough energy to carry it to the trash. Incidentally, Frank
is at B.C. for a while.
I am doing very well in the retirement home. That is to say I
have two bedrooms and a living room, and have arranged one
bedroom as an office. There I have my old desk and also a large
table on which I have a computer and my recorders (musical) and
associated electronic gadgetry. Since I felt I had to have something
definite to do, and had no interest in trying to continue research, I
decided to write a book just for the hell of it, not aimed at
publication, and picked measure theory as a topic which would
involve merely scut work. And one of my principles in writing was
that since I was in no hurry, in fact wanted to spend a maximum of
time, and had given away all my math books, I would work
everything out on my own. The project had the additional advantage
that it gave me an excuse to get a computer. It is a wonderful toy
and great time waster. I spend more time trying to get it to do what
I want than working on the math. The book is an introduction to
measure theory, written as I think such an introduction should be
written nowadays, with much more space than usual devoted to
discrete measures, multidimensional measures other than product
measures, examples from probability and so on. Of course there are








conditional expectations and a bit of martingale convergence. But
there is nothing really fancy. It is not written as a textbook. There
are no problems, and I have not even considered publishing. There is
more than usual about o algebras, but not a lot. My hardest problem
was to combine a reasonable amount on convergence of sequences of
measures with what is taught in probability courses involving
measures on the line. Everyone ought to know that a bounded
sequence of measures on a compact metric space has a convergent
subsequence, but that fact is hard to find in the usual texts. And the
relation between that fact and Helly's theorem for sequences of
monotone functions should be discussed. It is surprisingly messy to


cover this area.






101 West Windsor Road #1104
Urbana, Illinois 61801
doob@symcom.math.uiuc. edu
Rugust 10, 1994
Dear Chung:
I see by your address sticker that you have changed,
now only having two names. That shows you are willing to
change, and therefore I find it all the more surprising that
you refuse to advance to the end of this century and get
computerized.
What is your new book about?
I really have given up mathematics, and remember even
less than I used to, but I think I can answer your questions.
(Euerything that follows is in two dimensions.)
Of course every boundary point of a disk is regular, and by
Osgood-Caratheodory and inuariance of harmonicity under
conformal mapping it follows that every boundary point of a
simply connected Jordan domain is regular. Since regularity
is a local property it follows that if a Jordan arc A is part of
the boundary B of an open set then euery inner point of A at
strictly positive distance from B-H is a regular boundary
point. OK?
p. 815 of my book: what is wrong? The dates on p. 822
are funny but I think there was a publication delay. I think I
remember that the date of the volume was 1949.
Enclosed are some errata from MEASURE THEORY.








MEASURE THEORY CORRECTIONS

p. 12 line 9: delete 1 after the period.

p. 20 line after (3.1): The first a should be o0.

p. 25 (7.1): delete the second }.

p. 45 line 4: J" should be J'.

p. 52 line -3 should be ...ifandonlyifA* DA andthedifferenceA* -A has

p.64 The last two lines of the Proof paragraph should be replaced by

... are mutually independent, P{A AAk} > P{A}P{Ak}, and therefore (k -eoo}, P{A}P{A} = 0, as

was to be proved.
p. 64 line-1: Ae* should be A,.

p. 69 line 3: almost is spelled wrong.

p. 83 lines -7, -8, -9: f (without a subscript) occurs three times; each should be g.

p. 90 line 13: common is spelled wrong.

p. 92 (15.2): there should be a close parenthesis ) following S-G.

p. 118 line -14: T should be T--1.








Nov. 21, 1994
Dear Doob:
It was Borel who had the notion of Borel tribu (on the line) and the
measure defined on it beginning with the length of a bounded (ouch) interval.
The hierarchy of Borel sets through transfinite induction was well studied
by Baire and Sierpinski >.. (down to Dellacherie). It was Borel's grxax
idea of defining the measure of the rational by covering which led to Heine-
Borel's lemma later known as the definition of (bi)compactness. [In 1945
the 'biV was still commonly used.] That lemma is necessary to show the con-
sistency of his definition of the measure. Very. very few living mathemati-
cians know that Borel mas never x t~xxt provedthe existence of his measure by
the transfinite construction. The question we want to ask you is whether you
know such a proof anytime and by anybody? Please answer quickly to settle
a point of history.
I know that a young man (in 1956) gave such a proof which might have
been published (possibly as "logic"), but we are still tracking this doNn.
To my best memory he was called LeBlanc but this is not certain. He had
died long since. I had lunch with him in 1956-7. [I haven'* checked MR on LeB.
It is useless to ask people like the Bourbaki such historical questions.
It is well known they usually got the history all wrong, either owing to ig-
norance or owing to arrogance, most likely the two together. Many known ins-
tances such as Zorn's Lemma ,.... To show how "senile" some of these gents
had become (decades ago) I inclose a page for your "crossword" puzzle.
Bernard Bru (do you recall the name) has full access to the French ar-
chives, knows a lot about Bovel vis-a-vis Lebesg e etc. I am urging him to
cooperate oJa little essay on the true history of "Borel-Lebesgue measure".
You are author of a recent (most recent on the subject, I think) book on the
subject, old enough to kno", somne history, so it occurred to me to ask you.
By the ways who first had ,the idea of an outer-measure? It could be Carath4-
odory.

Sincerely, Ise J41 h


-postponed wing to Sunday&
t Acc. to i5ru. L. was a nasty type'l /, (,
^ 'T,







101 West Windsor Road #1104
Urbana, Illinois 61801
doob@symcom.math.uiuc.edu


November 28, 1994
Dear Chung:
I think Borel's idea was to define the length of an interval
union in the obvious way, then go in the obvious way to
countable intersections, and so on!!! But I admit I have never
checked his papers. I know there was a quarrel between him
and the Lebesguers on who got measure first. You might
consult Neueu.
In my opinion Borel did not know what a proof was. As
evidence look at his famous 1989 paper in which practically
every argument is ridiculous. Frechet tactfully wrote in 1915
about Borel's proof of the strong law of large numbers in the
Bernoulli case: "Borel's proof is excessively short. It omits
several intermediate arguments and assumes certain results
without proof." Have you ever seen the analysis of this paper
by a couple of authors, probably from NYU, (Nouikou? plus ?).
They go through the paper and point out all the absurdities. It
is not that the rigor is not up to modern standards; it is that
there is no attempt at rigor. For example, without saying that
he is making approximations, he proves the strong law of
large numbers in the Bernoulli case, using the normal
approximations in that context with zero error terms. [The
earliest proof of the strong law in the Bernoulli case that I
have seen is by Faber (Math. Ann. 1918)].
Have you tried looking at measure theory in the German
encyclopedia? or the French one, which I remember is pretty
good?
I found the following in some notes I made once:


H. Lebesgue Paris C.R. 132 (1901), p. 1825.







Integrale, Longueur, Hire = Annali di mat. (3) 7 (1902),
231-259 (Paris thesis).
Inner and outer L measure in n dim. [but see also
independent work of Uitali 1984, W.H. Young 1984, Young
& Young 1906] Riemann int. = boundedness and uniform
continuity. Bounded convergence theorem.
----- nn. Ecole Normale 35 (1918), 191-258.
Discussion of measure with Borel later (see B.
1919)
Borel --Rnn. Ecole Normale. 36 (1919), 71-91.
Discussion of measure with Lebesgue.
W. H. Young, Open sets and the theory of content, Proc.
London Math. Soc. Ser. 2, 2 (1984), 16-51
Inner and outer L measure in RN learned of
Lebesgue after his own work his "additive class"
is defined by Caratheodory's criterion but he does
not show countable additivity; his following paper is
on upper and lower R integrals, see criticism in
Enzyklo. Zoretti-Rosenthal p. 979 note 405.
W. H. Young, 6. Chisholm Young The theory of sets of
points. Cambridge 1986
Measure, apparently independent of Lebesgue.

I once came across a paper by Frechet (1915) which
seems to contain a proof of the Radon-Nikodym theorem long
before RDN. The paper is very long and it would be an awful
job to check his work.
I haue neuer heard of LeBlanc, and cannot believe anyone
really got measure by transfinite induction.

Greetings,











December 15. 1994


Dear Doob: t, ac;a I'

You do not know the history. As I recalled, correctly, LeBlanc

published his proof in 1956: Oanadian J. of Math: On the extension of

measure by the method of Borel Cunderline mine), pp. 516-523. I did not

know two Portugese had given that proof before, in 1944 and 45, named

Albuquerque and Neves, see the references in LeBlanc's paper. If you don't

have easy access to the library and WANT TO READ LaBlanc's paper, I can

make a copy and send it.

What you said about Borel's GREAT Rendiconti Palermo paper was the

6th or 7th time you told me, but you had forgotten that I told you I did

not need Frechet (and that jewish Bill-Boone Novikov who was here when I

came] to know Borel's mistakes. I read it circus 1940 but did not see

Frechesuntil in Princeton'. You did not seem to know that so far as

Borel's Normal Number Theorem is concerned* I think he had the correct proof

because only the general part of Borel-Cantelli is needed (no independence)

and for Bernoulli distribution AIL estimates were known (to DeMoivre !]

Did you know that Borel published (c. 1919) a supplement to that paper to

prove a special case of the dependence part of his "i. o." result (notation

MINE. adopted by Feller)? so that his other result of the SLLN for conti-

nued fractions was OK? Few folks even heard of the latter, but Kuzmin, Khin-

tchine, Levy followed up: I mean the continued fractions.

Now another historical question you should be able to answer. Who ever
even considered or tried to prove the existence of p' (t fdr t>0O before
Donald Austin did it in 1953 in Syracuse? under qi ij(t ? I said in my
reminiscences of Doeblin (you have the article) that -1id not seem to have
anything on that despite his customary cockiness. He has a paper An finite
state space in which of course the result follows from those for&t=O. In
your 1945 paper you ASSUMED (ahum) these derivatives exist (and,continuous?)
before your study the equations. 'eller imposed tremendously strong condi-
tions to derive them 4C his Vol. 1. But there may be other literature
which I "ignore" (French sense). hence the question.









.-- wxvuvs.-utn.I in tnat paper are (1) the mathematical result

[still largely unknown and hence unemployed by good analysts like Edrei

Who once consulted me) P(limsup En) =0 if EP(En)
(2) the RECOGNITION that this (and sometimes its converse [STRONG in the

sense that fO becomes 1]) CAN BE USED TO PROVE BIG RESULTS. [UNderline

mines not understood by.such as William Feller or you, I think]. I wonder

sometimes whether William's grudge against Emile had been occasioned by some
"o't a iob Or b6owyse
knid of personal resentment? The Bourbaki hated Borel precisely because

this: he held too much academic power then in France for those feeble

ypung normaliens, among whom perhaps as Lebesgue whom at first Borel
patronized (counter-proteged)' According to Brus Lebesgue used to say Borel's

theory of measure ... until he wanted to take the full credit. Oes choses

arrivent.








101 West Windsor Road #1104
Urbana, Illinois 61801
doob@symcom.matlhuiuc.edu


December 28, 1994
Dear Chung:
I am amazed that Borel's method was actually made rigorous,
but the Leblanc-Fox paper looks OK. I found it particularly
interesting because he uses lim inf, lim sup and lim of a sequence of
sets, as I do heavily in my Measure Theory book. The difference is
that I apply these concepts and the set metric determined by a
measure to get the Hahn-Kolmogorov extension theorem etc. whereas
they use transfinite induction.
I don't think I have been unfair to Borel. If I remember
correctly, the estimates Borel used in his 1909 paper were obtained
by using the Central Limit Theorem assuming 0 error in the CLT
approximation! I don't remember Borel's later paper and doubt that
I ever saw it. I guess but of course don't really know directly -
that Borel's 1909 paper had a big influence in spite of its sloppiness.
If a less well known writer had managed to get it published it might
have been scorned. (If I had not been well known, my papers would
have been refereed properly and some of my nonsense would not have
reached print.)
Your non-p.c. reference to Boone and Novikov is beyond me.
Boone was a fervent superstitious catholic, a descendant of Daniel
Boone, and the Soviet Novikov associated with Boone, religion and
ethnicity unknown to me, is not the NYU professor Novikoff, (I
spelled his name wrong in my last letter) who criticized Borel.








Perhaps I should add, in case there is any doubt, that I
myself am an R ED I rE D

*MI SE EZ of the UNZVPERSAL

L lt. I2D0lJ3 1and as such have officiated over three
(3) weddings of which two have lasted and produced children.
You are a pest to bring up old problems which I have quite
forgotten. Unlike you, I have retired and never think about math,
never go to math talks etc. I gave all my math books to the library
except those by Doob and whenever a new math book, such as the
Doeblin book, arrives I send it to the library. All I have at home are
my books and reprints of my papers, bound in 2 volumes. I
remember that Doeblin had a wrong proof of something or other,
sample function or maybe pij(.) properties for Markov chains with
finitely many states, and I suspect from your letter that you have not
looked at my TAMS 1942 paper on Markoff chains.
This is my all. If you send me a preprint of what you are
writing, I might dredge up more references or complaints from the
murky depths of my memory.


PS. I shall be in Florida January 8-30:
Sandalfoot Condominiums 1B3
671 East Gulf Drive
Sanibel Island
Florida 33957






101 West Windsor Road, #1104
Urbana, Illinois 61802-6697
doob@symcom.math.uiuc.edu


November 25, 1996
Dear Chung:
I see you are spreading the word about me and
probability here next week. I knew nothing about it
until Burkholder mentioned it. I have not yet decided
whether I like your title P&D; the reverse possibility D&P
- would imply that there is more to me than P. Of course
any embarrassing remarks might lead to my retelling the
tale of your encounter with an Ithaca turtle! If your
schedule is not too full, we must have a meal together
and I'll introduce you to my female connection, an
addition to my life as gratifying as it was unexpected.
I stil go out on the Saturday Hike; if you want to
come out again bring along suitable clothes, especially
footware suitable for mud.
That historical paper of mine in the Monthly is the
only paper I have euer written that was a joy to write.
It is fun to quote nonsense by others instead of
publishing my own.



^^-----







101 West Windsor Road, #1104
Urbana, Illinois 61802-6697
doob@symcom.mathuiiuc.edu

December 7, 1996
Dear Chung:


I assume you got the Herox OK. I enlarged the original
a bit to make it easier to read. It was a lot of fun quoting the
nonsense that various people had written.
If you look at pps. 76 and 462 of my potential theory book
you will see that reductions (reduites to foreigners) work the
same in potential theory as in martingale theory. In particular
iterated reduites yield crossing inequalities in probability.
The most natural crossings do not go from below one
positive constant to aboue a second larger one but from below a
positive superharmonic function [supermartingale] to aboue a
second larger superharmonic function [supermartingale], the
second one being larger in the strong sense that the difference
is a positive superharmonic function [supermartingale].
The intuitive meaning in the probability-classical potential
theory-content content is (p.655) [did you know this?]: reduction
of the harmonic function! on a set A is (the probability of hitting
A) and that iterated reduction successively on N then on B on B is
the probability of hitting R, and then hitting B. In the martingale
context this makes the application to crossings obvious. The
immediate application to martingale theory is to martingale
convergence and sample function limits. The immediate





application to potential theory is to fine topology limits, in
particular Fatou.
I still do not understand the following. The fundamental
convergence theorem in potential theory (p. 70), basic to
reductions, is that the inf of a family of positive superharmonic
functions is a function which coincides up to a polar set with a
superharmonic function. The best I could do in the continuous
parameter probability context (p. 477) is clumsy and I feel it
ought to have important applications, but so far I have come up
with nothing. Rs far as I know, no reader has noticed the
theorem.
When I wrote the book I thought that potential theory was
an important subject, too much so to be thought of as just
something not very interesting, whose results could be obtained
directly from probability theory, so I put potential theory first in
the book. But I had no real plan, and in fact half of what I wrote
was new to me had to be learned and translated into my own
contexts, but as I wrote I saw that potential theory and
martingale theory were really two aspects of a larger theory.
The reduction applications illustrate this.
Season' greetings to you and Lilly,






(2-17) 3G7-7c7

101 West Windsor Road, #1104
Urbana, Illinois 61802-6697
doob@ symcom.math. uiuc.edu


December 11, 1996


Dear Chung:


I was not clear and explicit enough in my letter. Also I forgot what I once
knew. Consider the following correspondence:
Classical Potential theory Discrete parameter Martingale
theory (finite or infinite sequence)
[super]harmonic function [super]martingale
A positive superhamonic function has an A positive supermartingale (infinite
a.e. boundary limit function, for a suitable sequence) has an ae. limit
boundary and the appropriate topology
(Martin boundary, fine topology limit)
Potential: a positive superharmonic Potential: a positive supermartingale
function with no with no positive martingale
positive harmonic minorant except 0) minorant except 0)
Riesz decomposition of a positive Riesz type decomposition of a
superharmonic function positive supermartingale
Quasi bounded harmonic function: the Quasi bounded martingale: the
difference between two harmonic functions difference between two martingales
each of which is a countable sum of each of which is a countable sum of
positive bounded harmonic functions; a bounded positive martingales; a
harmonic function is quasi bounded iff it martingale is quasi bounded iff it is
satisfies a certain uniformly integrability uniformly integrable
condition







a potential is represented by a certain a potential x. has Doob
measure representation as xo less a certain
positive sum (no Green function
analog)
A pos. superharm. function is a potential A positive supermartingale is a
iff it has boundary limit 0 in a certain L1 potential iff it has a.e. and L1 limit
sense. 0.
A harmonic function is a Dirichlet solution A martingale is the sequence of
iff it is quasi bounded. successive conditional expectations
of a r.v. iff the martingale is quasi
bounded.
Iterated reductions of a positive The corresponding iterated
superharmoic functionon a pair of sets reductions of positive
(p.76)have a convergent sum with a simple supermartingales yield the crossing
majorant inequalities.
Monotone increasing sequences of Ditto for supermartingales, but the
superharmonic functions are proof (Meyer or Doob) gets a bit
superharmonic, under the necessary tricky for the continuous parameter
finiteness condition on the limit function case and right continuity
A monotone decreasing sequence of A monotone decreasing sequence of
positive superharmonnic functions has a positive supermartingales has a
limit coinciding up to a polar set with a supermartingale limit, but the proof
superharmonic function. (Cartan, a deep and even the formulation get tricky
theorem) for the continuous parameter case
and right continuity


The iterated reduction arguments to are precisely the same in the two
theories and therefore in my potential theory book I give only the argument in the
harmonic function case.


-'I






Obviously the two subjectscan be subsumed under one larger theory but I
could not formulate such a theory in an elegant way when I wrote the book.
Classical potential theory is symmetric and therefore has no analogs of
the theorems for backward [super]martingales, but the potential theory for the heat
equation has such analogs.
Everything in your letter about Borel's 1909 paper is piffle. He is the guy
who thought that it was sufficient to define Lebesgue measure on the line for
countable unions of intervals and their countable intersections and then say "and so
on". In that 1909 paper he derived all probabilities by applying the central limit
theorem, making the error term 0. That is OK for a first elementary course on
probability maybe, but was not any more rigorous in 1909 than now.
I am surprised that Bernoulli had the inequality you quote., but to use it to
get the SLLN one needs the Cantelli half of Borel-Cantelli, because the events
involved are not independent, and Borel did not have that half.
Uspensky's book has lots of good material, although I quote his stupid but
then standard definition of probability in my Monthly historical paper. Most
probabilists in his day were OK as soon as they got away from the definition of
probability and reached the math. that probability contexts suggested.

Read Borel 1909 again.







Dec. 16, 1990


Dear Doob:
I just got yours of Dec. 11. This is the only letter I've
received since my trip to Urbana. Hence I don't know what you

meant by your first sentence there. If you sent me an earlier
one (after my trip, not before) please send a copy.
I don't think I saw your "Monthly historical paper" mentioned
on the 3rd page of your Dec. llth letter. If you have a reprint or
can make a copy please send it. Otherwise give me the exact refer-
ence though I don't care to look for such issues ...
I wrote you a second letter typed and mailed followed by a
postcard. It is not clear if you had received those when you wrote
the Dec. llth Since I did not ask for the tabulation you send, I
wonder what instigated the tabulation --- I said little about this
stuff because it must be well over the heads of most in the audience
myself NOT excluded. By popular demand (and approval from Frank)
I am drafting a modified version of my talk. I called your left-
wing buddy Chandler to offer it "as a vaor" to his journal. So
far he has not responded. But it can be publsihed in one of the
probability journals. Those left-wingers tend to be more dictatorial
than their counterparts, I learned long since. Too bad for the non-
probabilist readership.
All about Borel was in my typed letter. My only que tion now
6 is whether he knew how to write down HIS set --( L. If
/N \ -s h=kn V0
he did he got everything. My memory failed at this juncture though
I have his Rendiconti Palermo 1909 somewhere. Oheerio.




VLNoUU /I









101 West Windsor Road, #1104
Urbana, Illinois 61802-6697
doob@symcom math.uiuc.edu
December 19, 1996
Dear Chung:
The reference to my historical paper is:
The Development of Rigor in Mathematical Probability
(1900-1958), from Development of Mathematics 1988-
1958, edited by Jean-Paul Pier, Basel 1994, Birkh5user
Uerlag
I just copied this out of the Monthly reprint, which
gaWi;wae! Of course the book will not be as widely
distributed as the Monthly, so the reference to the
Monthly is more useful. But Why do you want the
reference? Don't tell me that you are really going to
publish your talk and that in it you want to refer to the
paper?
KI
.....--^,- z:f








101 West Windsor Road, #1104
Urbana, Illinois 61802-6697
doob @ symcom.math.uiuc.edu

Chung:

I think your point is that if Bernouilli had
only lived a little, say about 200 years, longer, he
might have noticed that his inequality combined
with Cantelli implied the SLLN. Too bad Borel
didn't see the possibilities. I congratulate you on
your memory. When I tried recently to read Doob
on reductions I found I not only could not follow
the reasoning but had even forgotten what some of
the words meant.
7,









101 West Windsor Road, #1104
Trbana, Illinois 61802-6697
doob@symcom.rnath.uiuc.edu
December 20, 1996

Chung:
I haue some fine references in my history paper on
the difficulty of getting people to appreciate newi ideas.
don't suppose that this difficulty is why UOU dislike
IIr COMPUTE ? I i uset think that the best ti-me
masters are fireplaces, but they are negligible compared
to computers. I spend hour after hour surfing the Web;
for exaump ,meas lust reading RLICE !N iiONERLiAND, a
beautiful iers.inn with the original illustration pliu a
musical accompaniment. Slightly more eriousir nno on

wiho has uiisd Ia ncmnuitr uinrd nrnrpssnr onuIld pepper
nant to go back to a tinnipwiriter. Alnd neither banks nor

uniersiitipes ncuid noui do their bookkeeping without
compnu ter. computer wmuld offer you the advantage
, to %%.,aG 's= an q, % w wjs %A %I ? % %A i M.. I a %j.%a ijM,
that the lettering contrast on a computer monitor is so
strong that you o"nould haue no difficult reading tent on
the monitor, even tent as small as this, which is the
standard size use except when writing to you.
IIIhy are y1o so. enamored of Borel? What has he
done for you lately? I admit that his idea of a countably
additive measure and its identification with probability
using dyadic f actions shows he had originality and
foresight buht he had onli sloppynn technique at his