John P. Walsh, 1996-2003

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5.s(b
we obtain (5.15)! except....
Except that Doob's result, Corollary to Theorem 4, is NOT applic-
able. It applies tp y(Qt+l), not y(Qt). That puny "1" makes a world
of difference, and it is t*hi$~iscussion of RANDOM TIME is all about.
A
Even though Qt is optional, we do not know if y(Qt) be haves like y(l)
or y(t); indeed we don't know if E(y(Qt)) is finite! In the very
special case where the common distribution is the exponential men-
tioned above (see below), this problem is a famous PARADOXe, pes
redoubtable Feller e-~ae h -ioutj. see [P]. In that case however we
can discover the true distribution of y(Qt) and therewith prove that
it has a finite mean. In the general case we are discussing the re-
sult seems unknown and so the argument above is invalid. Of courses
if we add a tiny extra hypothesis, for instance if the y's are domi-
nated by some big Y with E(Y) Renewal Theorem is true without such extraneous hypothesis and the
mathematician, unlike the practicing scientist, tends to be perfection-
ist.





Prof. Wafsh, Math UBC ee )
9 [se belW the foSt (aous
Changes to make in your NOTES SLbP OCo-S Cf by
164: though Fller. 'Y1V skhoL\, citeuit REA)
164, line -2: proofs hy ANOQas nY my M.C.
165: CITE and QUOTE (in his words) the page I faxed you months ago
about the lack of stationajty of revers. J'eh insite. He was
honest and dead. Many probabilists did not know that, you were
too young to know.
Aren't there too many Walshts referred to? Of. Part 1i
Sec. 6: No agasa and probably both learned from Hunt's discrete
t#me chains reversed from last exit times. I reversed CONT. TIME
chain (READ my M. C. book, 2nd ed.) T the first infinity TIME
b of the "minimal" subprocess that worried Peller and Doob so much
SDoob's ghostlike .turn to earth was viewed (before your time)
with the utmost alarm and mistruct. All that history was men-
C tioned in my notes in M. C. loc. cit. Quote it if you can.
< Nagasawa did well but certainly not before moi. Did you put
(is
my reprint on Martin bdy in your Ref? T~Inot in mine. DoAc~cSf
Sh-tryndforms: did you show an importantest case where the h is a
Martin bdy harmonic??? If not, try to add it as a suoerb examp-
le and refer to the Br. Motion case in ChungZhao X(FIND and
SREAD it, in a Lip domain). It is this kind of concrete cases
That most/ readers should be taught, not the general stuff.
Namely,that h-transform menas conditioned on X(T) --- did you
know?? I wonder if Doob did. PLt 1n iYur Pef-.
Sreduite: How about interior one? Brelot (with whom Doob meant to do
-o a book together) and some other "frogs" should be mentioned if
you know what they did. I invited Brelot here to give a course
but I forgot : there is his books) or ask somebody who knows.
Holes: your co-author cannot begin to understand waht you say there:
re-do it much more clearly wj proQtAesi
In case you got my fax ith Inq: you will mail her the entire MS with
all of my original text, Preface (one error corrected) AND REFs INCL.
You do your Refs. The INDEX should be combined: just add your new itemS
I sent you my chosen TITLE, and my part of the BEW preface. Am waiting.
r cREAD the first page + par paper and check the dats of the
d: ; references: Hunt 1960, Chung [4] 1196%. Japs 1964. Rewrite or just
/yc copy 9 whaj I wrote therefor SORRECT history. Pleass add my [4] in
our paper to your Refs --- I,unlike another author did not put it in
my Ref. [Both will be in new book(] perhaps because Chains are not
in the Lectures, now they are in the new book.





Prof. Wa/ah. Math UBCt
( tIe. beloa r.k most Afhmous
Changes to make in your NOTES SqvLbproeVs PoWptat by,
164: though Ffer, .y f ctet RE6AZy
164, line -2: proofs a y h s e n my M.C.I
165: CITE and QUOTE (in his words) the page I faxed you months ago
about the lack of stationary of reverse. J'eb inside. He was
honest and dead. Many probabilists did not know that, you were
too young to know.
Aren't there too many Walshts referred to? Cf. Part 1.
Sec. 6: Noagasa a~nd probably both learned from Hunt's discrete
Stme chains reversed from last exit times.* I reversed CONT. TIME
7 chain (READ my M. C. book, 2nd ed.) To, the first infinity TIME
-. of the "minimal" subproceaesthat worried Peller and Doob so much
f Doob's ghostlike uturn to earth was viewed (before your time)
'- with the utmost alarm and mistruct. All that history was men-
S ttioned in my notes in M. C. loc. cit. Quote it if you can.
Nagasawa did well but certainly not before moi. Did you put s
my reprint on Martin bdy in your Ref? Titnot in mine. Doafdw
A
h-trandforms: did you show an importantest case where the h is a
SMartin bdy harmonic??? If not; try to add it as a superb examp-
lQ Ie and refer to the Br. Motion case in Chung4Zhao (PIND and
READ its in a Lip domain- It is this kind of concrete cases
That most/ readers should be taught not the general stuff.
< Namely,that h-transform menas conditioned on X(T) --- did you
know?? I wonder if Doob did. .4 aut I fYouy Qfe j
Sreduite: How about interior one? Brelot (wi+.h v hom Doob meant to do
a book together) and some other "frogs'" should be mentioned if
you know what they did. I invited Brelot here to give a course
but I forgot : there is his books) or ask somebody who knows.
Holes: your co-author cannot begin to understand waht you say there:
re-do it much more clearly w: r pra+,Aes,
In case you got my fax with InQ: you will mail her the entire MS with
all of my original text, Preface (one error corrected) AND REPs INCL.
You do your Refa. The INDEX should be combined: just add your new item
I sent you my chosen TITLE, and my part of the BEW preface. Am waiting.
READ the first page p ptr paper and check the dates of the
references: Hunt 1960, Chung [4] 11960, Japs 1964. Rewrite or just
Scopy O f whaj I wrote therefor eORRECT history. Pleas.a add my [4 in
Y0441-
e our paper to your Refs --- I,unlike another author, did not pu" it in
my Ref. [Both will be in new book(] perhaps because Chains are not
in the Lectures, now they are in the new book.




COLE POLYTECHNIQUE FEDERAL DE LAUSANNE
EIDG. TECHNISCHE HOCHSCHULE LAUSANNE
POLITECNICO FEDERAL DI LOSANNA

D6partement de math6matiques


1007 Lausanne
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Dec. 11. 1996
Dear John:
Your were such a good letter-writer that I thought I would try
again, with several disparq requested
(1) At a recent reception a young^,aeme wh name I did not catch
(He...) told me that you had proved something with two renexl sequ-
ences. Can you send a preprint?
t (2) Thanks for reminding me of the Feller*Orey Fourier proof of
the Erdds-Pollad-Peller theorem. I looked at it but forget whether
the case Q is special --- if soa no good. A few youngsters have also
given a "new" proof using coupling, but theirs definitely made the Cb
case more difficult --- therefore not worth it. As you wall know, in
the Erd's proof 0 is the easier case If you did not know, read what
Landau said about Erdbv's modf/p a-0 theorem. I saw Halberstamwthe rec
ttjiea topand he seemed to remember.X I also quizzed him about Hardy e'
overlookiAg the genius of Erd8s -- the only "excuse being Hardy
was quite old already when he met the young E. Surely E. was in the
l Al sse Ramarujans and probably in his total output surpassed the In-
dian. Agree?
(3) What I mow seek is a brand new way of the E--P- theorem with
out special argument for the 0 case. It does not exit.
(4) Did that Chinese convince you that his proof of that David-
-Pn result is correct? But I am sure nobody knows how short it can
be made. Read Dyson's brilliant short proof of the a-1 theorem.
(5) I will publish a persoril obituary for Paul Erdba. I am
also writing an article : "Probabilty And Doob" which intervened un-
expectedly I O+-' year /41~f e
(6) I believe you took me to the high table at (xfor4 Jesus Col-
lege/
Anyway do yoL know Womersley at that colleagues His little gpa4 book
in the mini-Penbuin series sL know my GibblnRs but that is a snippet
from the tome] contained a fantastic mistake. I should like to en-
TM lighten~B(and also ask another question re Gibbon) but knowing some
of those Britishers I should be vexed not to receive a response. If
you can make him bahave properly I shall send my communications) via
Syou. Let me know soon. Best wishes for Happy New Year!


P. S. I do not do computer but do receive e-mail (too frequently)
with probadlity 1-e C[Littlewoddi 53. but for any important matter
a letter salould be sent simultaneously, cI
(7) for a few hours I thought the Austin proof of ury the
prettiest heorem ai'n ALL Markov chain theory can be somewhat made
more intuive by not invoking Lebesgue's monotone function theorem
Cin your as well my bool s but it failed. If you are qssSL interest-
ed let me know so that with your collaboration we may yet do it.



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THE UNIVERSITY OF BRITISH COLUMBIA
2075 WESBROOK PLACE
VANCOUVER, B.C., CANADA
V6T 1W5


DEPARTMENT OF MATHEMATICS


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Wmmnia@hotmail,com


From : "John Klmmel"
To:
Subject: PublIcat ti
Date: Fri, 7 Ftb. 2003 08:37:44 -0800
Deor Farid:


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Prof. John Walsh
A
Dear John:
Here is my initial reaction after glancing thru the first 6
pages.
Faites cannot be rendered into facts, so better just use
Englsih such as Generalities (same spelling in French exceot for an
accent aigu). Now you say thstsection is all about separability.
Look at the Index of my book and see what I said tb@4V For ex. I
am not sure I men ioed "standard modification". If so how will the
poor reader (remember$ a student not a professor of sorts) understand
what you mean there? How clev er that agini function is separable'
But that's what I found out, noyour acher Read my article "ProbH
ability and Doob"7for more details including a note on Wolfowitz,
whose son Paul may wa get you and me killed by terrorists. No jokei
since separability was not in my Lectures (except ip homework, see
Index)o If you need it you better explain it.
Next, "maasuarble process". See Index for Borel measurable: Sf
they are the same, why not refer ?/You can use "stopping" and need
not make high tech. wBrk too hard, why not just say I used another
adjective originallyy your teacher). By the way did you define
your stopping sans tribt? Better be precise.
I had my first chuckle when I read you made such abig deal
about Hunt's Lemma. Read Th. 9.4.8 in my Course, supposed to be
prerequisite to my Lectures. Eaysprof".
Is adaptedj ever defined --- not used by me. "Resolvent" is
mentioned but not used by me. Not enough to say your U is yxx
Ua IN THE BOOK elsewhere. Since you Ray this and that, you must
tell the poor reader tn t why resolvents in lieu of my previous semi-
group,"must be" used? I remember I said something therevaais ;t,,
The tex(merde)may help you later but it will save the environ-
ment if you pay some attention to other parts of the same] book
when you must use Frenchy words. Paul-Andre is dead.
A final and serious cpnseii du style : on p. 3, Iseparable:
"by looking at ...". On fa t tell choses debout vant le1 ort (?)
accompagn4 by hand-waving, but it does not mean a thing.' 9za
I prefer fax to phone, for the record etc. Owing to telemar-
keting, phone callsW are usually ignored: attention.

/ d ^ I dE f
d / \/cxa
I






2/18/2003
Prof. John B. Walsh [There is a John WalshLhow ,I have mot

Section 2. Def of supermedian, 1-, 2-, ... ? J Ke .
Did you ever tell the student (') why Ray and not Hunt process as in
the first part of the SAME BOOK? And why Branching prhnts? Where do
they evr bug-ff? How come Gil did not want them and what have they to
do with electricity or finance?
A
Preditable time --- remind me what 4ourf jpint paper about Meyer's
theorem is all about? it Ci I- Z f.i
p. 14 and before: Feller prcerty)f (see my Index)
15: S-S I don't accept this notation
16: StonpW- The rem (you have the Theorem sometimes): I do not
expect MY students to know what it is. Refer to some book in the Bibil.
Do you really need the fancy result? Wouldn't di. alone do A?
p. 17 and eaier: before any talk exhinit a non-stupid example
gm of a branching point which really shows up. Not the kind of/
Ckw junk Blumenthal used to cite. That does not work.
I invented Moderate MP, ~iA t bugged me when a Markov CHAIN path
'I i atsn when a
Sy hits a "boundary". w ad this virgin remember it.
,lt p. 20, I would delete "simply" an EXPLAIN the next sentence
Where is "inaccessible" defined?? Who knows what it means. ahs eal
Prop. 2.3.: isn't this just MY DEF of MMP?
S Polar --- refer to old text'. Q Mm
Sec. 2.4: must say martingalee on the SAME triple "otherwise it
-doesn't make any sense./ p. 22, Lemma: k is misprimt
p. 25: You expect the student to know Prohorov's th. I used to
Know but never, never needed it. h0llc JlIo I (a 6V
p. 24 Refer'Upcrossihg lemma to MY COURSE unless it is in
Sthe booked)
p. 26 (T-t) needs brackets --r T A/u4 /0 4
SNow I wonder if all that jazz applies to L6vy proqc If not
S what's al f&ss for, Hunt wrote his memoirs to APPLY to Potentials.
He failed in his second memoir because no reverse/dual which led
t~ Blu-Get's'duality hyp. Now if you have "Time Symmetry" can you
CIO do better? otherwise why all thertpreliminaries? Ca fCPJ a/ cO*3
.- p. 28 How about"a few"6rds're excessve/supermedian/.C.F.
Prop. 2.8.2 Isn't this just Cartan-Doob- ME (I have a short
version in Meyer's SeWinaire --- check it?)'
0 r'-9-LR a 4 c^ ^u ^1/pL t^^Lf^L^I
Cl- 1 1\ i-/_>; ,.





TRANSMISSION VERIFICATION REPORT

TIME : 02/10/2003 07:10

t foY- J n A YAl? l EL UR
p. 30 What's Ray's flawr sa i
Dh[E,TIMe 02/10 07:10
FA N /NAME 1 108341 065430397
DUR ON 0:
P 01
SULT OK
MODE e I 'o tP ST DA


Here are random notes on scanning your MS. I sent yojsome already

P. 30: explain Ray's flaw? -
P. 31, I never mention Lusin : what is it? Why do we need it?
Did yoi know Luz as accused of counterrevolutionary by Kolmogoroff
et al. and was barely saved his life by STalin.
Where is the BEEF? That's the main question. I have forgotten
all about Ray etc. and my text is all about Hunt process and more spe-
cial- ones never, never even the silly "stand ard" one. Thus for the
student of the new book there isa quantum leap which the book must be-
ware and prepare him for. -I see few signs af this. You are just ad-
dressing a new crowd. For those like US (many) who want to learn pro-
bability for possible applications to potential (Doob, Hunt, even
the dead Meyer) the first t irty/forty pages look like just preparing
the dough -- so, once more: WHERE ISTHE BEEF? You have written it
and of couse can keep it*,but you certainly owe the beefeaters a ot,
repaeat:a lot, of explanations to help them see the "relationship"
(these days on TV shows it means SEX --- the realthing') between
j1(y Co acht4;'2 *- *fcyWc/V2
all these new ro1olin OVa und foreplay to be recommended by Phil,
S that taEdNo. 1 show on Channel 7 here, surely also on yours)
and the EAL THING that Doob, Hunt, et el engaged in and exposed in
my original text. Take a peep now: there is fun there too.
p. 38, Prop. 2.11: what is E, and F? p. 37,Is "useless" a math.
BEF. If so where is it defined? p. 48, Th. 6, ba6 most of your
props have three numerals: is it not confusing to havesay31.2.3, then 67
p. 53: "time homog." Statt. trans. prob. Choose ONE. explainn 1
esy' double reversal --- may be easy for John W., not for oor student.
EX. in (3.9) you have het and tilde('a? why both in same formula.
I have forgotten what's what. w; pj hen I get time I will try the
crunch on -time reversal where I seem t recognize my old symbolism
/, I don't promise I can read it altho' I recall even that jerk
David Williams in his book "praised" MY style of (position.
Let's not spoil it via Ray etc. 6-s!A )





URGENT
Prof. John B. Walsh, Math, UBC

End of February, 2003

SDid you get my comments, two pages sent twice separal ?
More random notes while scanning (only) your script:
p. 30what is Ray's flaw (he was my teaching assistant)
p. 31: Define Lusin space and EXPLAIN why we need this
generality. Even Bourbaki claimed to be "moderately general. y
MAIN QUESTION for the first 30/40 pages: WHERE IS THE
BEEF?
My book is all about Hunt process and more special ones. You
can generalize as much as you see fit, but you must prepare the
student (not professional) for the quantum jump from mine to
yours. I see few signs of this. Too lazy to look at my book and
A
too messy to explain the raison-etre of your preliminaries (fait-
es diverse et pourquoi9J Your teacher Doob, my friend Hunt,
even the dead Meyer, did the processes to apply (MATHEMATICALLY
and ahysjcally) to potentials. As I told you last time, Gil
had to restrict, specialize his hypotheses in his second grand
memoir to do some of this, later further restricted by miser-
able duality hypotheses in Blu-Get. Maybe you have done better
by yout time symmetry? If so tell the student where all this
fooling around with spaces and compactifications and resolvents
come in? Sounds like a lot of foreplay (sure, recommended by
v~. talk show hostess gprah and host Phil: have you watch them late-
ly? I did not.) without the REAL THING. Do it alright, but try,
repeat: try, to motivate it/them a lot,lot more and tell the
frustrated unfulfilled audience why and how it/they help the
RELATIONSHIP [nowadays I can't turn on the radio/TV for one
minute without hearing this word three times plus another
three "INCREDIBLE" ]. Take a longer peep into my book and
see how your foreplay can fit in, or else skipped? You owe
the student at least a few more references: use my Index.
p.38, where is E, where'is F defined in Prop. 2.11? p. 37: is "useless"
a math term? p.48, how come Th. 6 does not have 3 numerals as other
propositions? I can't find Th. 5,4,3, ... P.53 "Time homog." is "Stat.
trans. prob." SAY it openly. If that double reversal is so easy* SHOW
it. This reader is totally confused with the double use of hat and
twiddle, e/ g/ why 1has ', etc. etc. I'LL stop here and wait for
a return fax: do not phone. se e. e.
-(3-, -C

















398 I SUPPLEMENT: MEASURE AND INTEGRAL


(iv) Let An E 9, An C An+i for all n and A = UnAn. Then we have

(35) limE(A,; f) = E(A; f).
n
PROOF. Denote f by (33), so that 1Af = j bjlABj. By (i),

E(A; f)= bjy(AB,)
j
with a similar equation where A is replaced by An. Since A.(AnBj) Lt(ABj)
as n f oo, and 7=1 t E l as m f oo, (35) follows by the double limit
theorem.
Consider now an increasing sequence (fn} of basic functions, namely,
fn < fn+i for all n. Then f = limn t fn exists and'f e S, but of course f
need not be basic; and its integral has yet to be defined. By property (ii), the
numerical sequence E(fn) is increasing and so limn 4 E(fn) exists, possibly
equal to +oo. It is tempting to define E(f) to be that limit, but we need the
following result to legitimize the idea.

Theorem 8. Let { fn and {gn be two increasing sequences of basic func-
tions such that

(36) lim 1 fn = lim f gn
n n
(everywhere in 02). Then we have

(37) lim t E(f n)= lim E(gn).
n n
PROOF. Denote the common limit function in (36) byf and put

A = (C e 02: f(w) > 0},

then A E -. Since 0 < gn < f, we have 1Acgn = 0 identically; hence by prop-
erty (iii):

(38) E(gn) = E(A; gn) + E(AC; gn) = E(A;gn ).

Fix an n and put for each k E N:

Ak = o S2: :fk () > gn() .

Since fk I fk+1, we have Ak C Ak+1 for all k. We are going to prove that

(39) UAk=A.
k=l





(60ou; (o07z
URGENT: Prof. John D. Walsh, Math UBC
Re iVMarch 4, 2003

Dear John


Haven't heard from you (fax, don't phone.) since I sent
you successively tree fax(es) of my reading notes. The
last for identification purpose)contains the KEY: FOREPLAY.
Now I FINISHED the whole bunch. My comments are meant
to S43e-s-tL
only to be suggestive, i.e., many more to be otedin like
A ( osi'fer&!
mode by the author yourself. It is not exhaustive_ _
p. 60, line 4: "since ..." I do not see.
.%/ (-t)
p. 61, 1. -8: better "...for which t it is rational ..."
A


p. 62, line 1:


I am utterly lost in all the U with and without


hat, twiddle ...


By the way is there one


without any of these? "IjS ces nofef,'-, i'
p. 63, 1. 8: I assume you defined before mu-polar? ire]
Hunt's Lemma --- did you not make a big fuss about it and
it is in my "elementary" text, I to/ y'tl.
p. 65, Prop. 3.2.1: "compatible" defined where?
Th. 3.0.1 --- such a BIG THEBREM with THREE numbers? SO how
come you had also some (not so big) Th. with just one number, as


I noticed and warned you about in an early fax ?


What is your


numbering system? Even Bourbaki worried about theirs.
p. 67: I can't distinguish the two Tis there. CaQ) yogr2

j0 I seem to recognize a bit something in the reversing
theorem, at least that small "el" j used. Do you know I did
it with the same "el" in my MC bbok, where reversing from
the first infinity was carried out? FOR THE FIRST TIME IN
MATH (cf. my sly historical comment on dead Meyer's earlier
effort in the book that earned me a good piece of canadian
bacont. But now you cover up everything with so many
hats, tild4s, top-dashes, ... I could barely see the forest.
Let us all hope in the nex Chapter some real screwing can be
viewed? Maybe even a little advancement over the -ew re-
sults in my Lectures where reversal/duality is visibly CwJ )
exposed?_ Some such thing poor Hunt his pupils, the japs,
... tried#to follow up? By the way, how about asking A
Dellacherietread parts of YOUR book, he is alive isn't he?
,Q u Your other pal Well is "useless" (ake your undefined set),
4U1BC1 but maybe that other probabilist(!i-Caw Ym6ght ) can help?
Add has / y I 4f' can't owing to age. Knight would like to see his
IqooK name there but I wonder if he will really read it? W
k LC / S/ A;mn



















3 MEASURES IN R 1 393


If we intersect both sides of (26) with the complement of (ak, bk), we obtain


[b', b] C U(aj, b').
j=1
i-k
Here the number of intervals on the right side is I 1; hence by the induction
hypothesis we have


F(b) F(bk) < (F(b') F(aj)).
j=I
jAk
Adding this to (28) we obtain (27), and the induction is complete. It follows
from (27) and (25) that


F(b) F(a) < (F(b) F(aj)) + e.
j=1

Beware that the I above depends on E. However, if we change 1 to oo (back to
infinity!) then the infinite series of course does not depend on E. Therefore we
can let E 0 to obtain (21) when the "=" there is changed to <", namely
the other half of Borel's lemma, for finite a and b.
It remains to treat the case a = -oo and/or b = +oo. Let -
00
(-oo, b] C U(aj, bj].
j=1
Then for any a in (-oo, b), (21) holds with "=" replaced by "<". Letting
a -+ -oo we obtain the desired result. The case b = +oo is similar. Q.E.D.
In the following, all I with subscripts denote intervals of the shape (a, b];
7 denotes union of disjoint sets. Let B E So; Bj E So, j E N. Thus
n ni
B= li; Bj = Ijk-.
i=1 k=1
Suppose
B=00Bj

j=l

so that
n oo nj
(29) EIi = Ijk
i=1 j=1 k=l





(0 0) f bl0Fq
Prof. John B. Walshf March 7, 2003: 7:15 AM
Let me hope your )ax is not BUSYjNO RESPONSE at this hour,
0 'n to-d6ow I am sure you wont get b=t5 ti -
RELATIONSHP. RELATIONSHIP, RELATIONSHIP. /^ )
Call my old text A amd your forthcoming B. Not in Preface,
NO, but in the Introduction you must tell the student why you needt
spend 50 some pages on all that jazz: lesolverts Compactication,
-Branching,..., EVEN REVERSING [TIME SYMMETRY in the title!] to DO
or OVERDO what is already done in Part A, without any of that junk.
After all, Hunt did not need Ray etc. Exhit as many as you can
easy-tS-see THEOREMS covering more ground than those in Part A/
And TELL IT not only HOW but also WHY Part B is worthy sequence
(as printed) to Part A. The buyer should wonder.
By the way, take a peep into my INDEX to see what Part B
will RELATE to Part A. I saw those symbols
( A
-.~

ARE they going to be expanded in Part B? Remember "nearly Borel"
A
---have you Ray it away, and what about Section/Projection/Capa-
city? As I told you I can't read it anymore so .the ball is in
your Park()' Out fax worked perfectly" (650)857-020.on
Sabbath too.





s At C -

3ik J^u^^.








0 Saturday Marc# 8, 2003
i Office closed, No Fax
SHey here is where your BRACHING should be illustrated. Take a
A
q look at MY MC book, 2nd ed. where I not only reversed the path
From T/ but introduced that fatal el that will now appear
in another 2nd ed. Now if you have a copy of my Acta Math..
xx 1962 [did you exist then?], MMP was discovered at the last
stage of page-proofing. 40t/ years is a long time but I seem
to remember my futile struggle to get SMP at that "Branching"
and nearly faulted. Grace a la den're epreuVe* it suddenly
V- A /C jumped to my mind: MODERATE basta [no Frenchy or Anglo word
ap proysMNGes A
Nearly ipno this great Italian bon mot 1. Your blow by
blow notes on my three earlier notes cheered me enough to
plan a "SANS BLAGUE" (pre)historical epilogue to the section
in Part B (perhaps also a hint in Part A) where those miser-
in PartB B a
able branching took so much space and time. You should EX-
A^ A A<-'-.
i PLAIN how that first "explosion" (Feller-Doob) and the tri-
E vial special case of ghostlike return to earth (my words,
Your teacher's "stunning" performance), as well as the COM-
IPLETE picture described in my Boundary book long sold
ouI g-L Andre cooper F Par example: when the path
cpmes down frm ao (a boundary point) either stickly or
nonstickily (fee my Bpundary book), is IT still a branch*
ing Apointi? I do not know but you should spend six Itx
seven lines TELLING the reader what's up. This is
what I call a REAL example not those "uniform motion"9ye
A junk. I am typing thisas well for my own MEMO when I
-;x(I my Part A referring to Part B to be filled in by yo

^ A et& 4AC p n v-u








Sunday March 9? ,2003


Dear John:

I am inclosing two old reprints you may not have. I assume y&u
have reprints of our paper on reversing. Iust read the first few para-
graphs (recall David Williams in HIS BOOK complimented on its clarity]
If your Part B will contain most if not all the stuff there (too many

pages), at least you shoudl give some "APPLICATIONS" --- of course Mathe-
matical ones if not physical. Otherwise why bother? tay think so.
Re my old reprint on Hunt's Hypotheste, this seems relevant to th4
first fourty-fifty pag es of Part Bt much improved of course? But let me
know if the results in Part B have anything to do with wyat I did there.

I gave the talk in Rome and on my return stopped overnight at Ovieto, danK
the marvelous Orvieto AMABILEarecommended by the vineyard owner who is(L
_t nor on v' USA & ca nada)
also the owner of the rand hotel I stayed in and who came to my table
to greet me. Those were good days. My ntop was at Strasbourg with the
Walshes --- did we eat at *Zimmer', (maybe not with you).
-.r -eve. Cro c.< letx
I did not thlAk I would look at those old papers again, most of
which I do not understand now. Caest la GUERRE.
I have only begun to read your response to my "notes", do not
Expect to understand all of it and will not make further suggestions.

Tdg(t follow those Frenchy and probably did not ever use words like
adaptedd". If it is not in the Index then you must define it, ditto for
other terms. Meyer is dead anyway By the way/ are you citing their
encyclopaedia anywhere and is Dellacherie still in Rouen? I alwefs want-
ed to see Rouen* By the way, are you going to France this summer? Those

4udes will probably start the war when you get this letter; some patriots
are already boycotting French wines, but you are Irish/Canadian. Tell yoty
resigned minister that "moron" seems appropriate. Ciao. L


s)ir i'4erc? C1/AY /Ae(- a5k 6h "* /-0 0PrrT-.-






3/12/2003
Prof. John B. Walsh

I sent you a batch via IsoJLe which will take longer time. I've
begun marking stdthe numerous minor corrections in my old copy, and
found out things you may want to refer to. On p. 136, notes on
Sec. 3.8, Ray and standard are mentioned, so now may have to refer
forward to where you define those, if I remember Y More import-
ant for the reader is for you in your Part II (I changed A and B to I
and II) to tell him where you go from Ray etc. backward to my Hunt.
If and when you do serious theorems in a more general context than I
did, it is your duty to tell the reader .... A
It may help the reader to print in large letters a Notice like:
From here on we will deal with a MP with the following properti-
es: (A) (B) (C) (C).... Cf. my list of properties for BM bt>
fore its special results. e I,, .
Both "Adaptedtkand PStone-Weierstrass" are in my Course, see Index.
A special form of the latter is spelled out and sufftces there. By
the way if yourUBC is not bankrupt you should tell it to BUY a
copy or twoTthe njENLARGED edition of my Curse with complete
exposition /f me'er Rz ,oi)
exposition f mear theory. I have a couple of ccopies for
sale (with discount.) to anybody who can afford it.
Fin4 HYpothese (L) and (B) in Part X (formerly Part A) and
let's see if you can get away without them? Does not your junior
college Ed Perkins(ii& Barlow) have the capacity to really read
the new math in your Part 3 and possibly find errors and make
improvements? I can't. Fax any reply, No phone.
USA(650)857-0208

L'L-C




;OLF naO71b








URGENT: John B. Walsh
March 25, 2003

This is the second fax sent you to-day, while you are
sleeping. The first is to ask you to rediscover Boob's no-
Projection proof of Meyer's virginal effort. This second ona
is a slightly less old nut, Theorem 1 in my Stopped Feynman-
Kac functional, Seminaire XIV (1978) Strasbourg, re printed
in the Second Edition of my Green, Brown, and Probability-,t/'Z
p. 166 Footnote. Sine our new book has "time Symmetry" in
the tttle, we should give a new proof of this obviously time-
symmetric result. I've prosed this venture at several inter-
national gatherings, the latest maybe at Cornell where that
Russion guy who wrote papers a yours (name forgotten) was
A
and I challenged him X*B-that. Zero answer. Hope with all your
a paratus in Part II you can do it. It would be a true bench4
mark". It is true and proved (by me in another paper) for all
one-dim Markov process with continuous paths, sometime misname
ed "diffusion". [By the way if you use that term you better
define it.] Once it is probed by reversing the TIME, I am
sure you can extend it to more general processes. By the
way the function q there can easily be generalized to integrab
rable etc (I assumed it to be bounded).
Let's see if all that jazz can Ke used to kill a fly?
Bonne Chance! Why not try it on some youngsters like
your colleagues at UBC? 4


I have copies for sale (at a disocunt) to your libra-
ry or other customers. Is the First Ed. in your library?
The 2nd has a lot more stuff.






( Dq-)


URGENT Prof. John B. Walsh


0?


Hey please sketch (not too briefly) that famous proof on that
flight from Zurich to Beijing and send it to me for possible rewriting.
I want to add it to that anecdote in the Notes so that space is avail-
able without pain. Your teacher published it somewhere but I am sure
RD A


you can recAver it. That will be great fun in the
also Hawkes's remark about duality.,,CrIreversal.


new ealtion. au
Can you show some
nri* ^ r Tim


now with your big buildup?? Bonne chance. Reply by ^ o-

by^i (Atc


saUxMH f




1.-


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3/24/2003




URGENT: John B. Walsh


March 26, 2003
Five pages about M. C. are here, in fact some duplicates came
earlier. This revue is an excellent idea. The only "probleme" [the
English version of this good French word is being so "incredibly"
(another fantastically abused word in the current American speech)
abused that I hesi te to EMPLOY it, but the French word means what
I really mean. --- OK the only problem is who among the readers
can appreciate your Examples without a rather deep acquaintance we
MY (hey) two books on the subject. Let me suggest a few particular
REFERENCES to help those few readers, later. For the moment, I as-
sume you are working on the two Math Problems [correct usgge here]
in my last two faxes, one in each. The second one as an Ex. of re-
versing may be too hard even for you. [That Russian said it was
"unfair"! What's his name who was at Cornell then, c. 1994.] So
let me just mention the first about poor Andre's theorem. I trie
ed to read it just now but had to quit. But please tell me:
(1) That fancy projection around p. 40 of Part I, using
analytic set and all that (forget about Doob's airborne
dismissal of all that stuff): are you using such advance
ced weapons in Part II? If so yoL better say so.
(2) Assuming you learned all that stuff in Strasbourg, can
you tell me why Meyer told me c. 1983 in a letter:
(i) progressive measurability is unimportant.
(ii) that theorem should be called "section" rather
than "projection"? If you can tell me I may
change the name there (in Part I) though it
is too late for Andrd. How many times is it
used in PartII? If you do not do analytic
sets are you just going to do as I did by
handwaving?
As you see serious questions of prerequisites for the new ed.
may be involved here, since Part II seems on a higher level of
generalization/abstraction than Part I intends C'est la gue-
rre. rtotbe)

VL

Ckc





6 y- < -2Z 0';7
Not.too urgent: Prof. Walsh M
March 27, 2003


Dear John:
Re M. C. you should say a few words abotu the differentiabi-
lity ,f Pij(t), about qi= oo, the case of "conservative Q", all of
which may be referred to my book and some may be more ready in my
Boundary book. Meyer did a great injustice (somehow I did not catch
it then) in calling the diff. th. Ornsteinl, It Austin-Ornstein. 0.
proved the hader general case which however is (till now) useless. It
A
is Austin who proved first the useful (for your Part II') case but my
later integral representation may be more relevant for aum purposes.
Anyway unless your part has a lot of diff. eqs (I have not seen any)
that thapter'of differential equations should be motivated. Bru sent
me the big MS by Doblin (the way he wanted his name spelled) now al-
ready prinj (see his mention of Austin I t4ld him: Yincent D. never
did prove that! Curious. Nor Kol.eV al. It was On my AFOSR pro-
ject'r0 t I hired Don and he surprised me by proving these thinSC.
(Doob did not, could not). Take a look at my MC book and add a few
historical words in the Notes [there's where I amuse$ myself. I
will state my reversible theorem for F-K functional posed to you:
that Russ was a student of Dynkin's who hid behind his master at
Cornell but I did t6Q1 him the problem. Ydu do not know his work
on reversing etc? A long paper he sent me but I forget even his
name: ask someone else). About your lost MS it was the memeo
S4 copy of your talks in Paris I think. It should be relevant to II,
4 i6- how about your published article having to do with last exit ---
I forget the theme but expect it has to do with II. Anyway we
i e0 should finish by June whatever you add. Y subtract. As I realized
l I can only make minor corrections in Part I, add a few cool remarks
nhy ? [Eggd ftr.Hawkes to say so, do you know him?] and LEAVE THE
REST to Part II and you. You can also say whatever you d)n't ROV(
in the Notes for your chapters. Are you having the math checked
by someone in UBC, and how about Salisbury? (I ; ~




(C





URGENT: Prof. John Walsh


I sent you yesterday my comments on ydur Examples with Chains. %
Here are a few more page by page marginal notes/ J htst s&kp MOL,
2, before Kol equs say that EVEN the qij's requii* good work by
Kol. of course Ref. to my book. Now tell why we need those stupid PDE?
p. 3: Ex. 1i when the path reaches co, I WOULD remind the student
that do is NOT a lumber hence out of'space. RefeF then to your teacher's
ghostlike return from there yonder: of course in my book. A whole section
is devoted to the minimal chain there. Unfortunately when boundary points
etc. are concerned, Ref. to my other book is required. A few specific
page or section references should be given. yralLca' / e e"
If all qi=-qii are finite, we have right continuity (proved in the
2nd book actually first noted by Pittenger). I forget what we can say
abour the rest of Hunt hyp.? Maybe you should say p.3 p. 4: Are you
going to tell how boundary POINTS can be defined ( o Ray et al.) How about
nonsticky bdy pt and explain why a sticky bdy pt does not branch? Hope
y6u will spare the pain of Martin bdy?? Even that is mentioned in my first
book, I think. 5 fte sllOdj bra~nkC
p. 4,lower paragraph: I don't know if Yt = f(Xt) is Hunt when Xt
is: surely we need some condition f? This is the.kind of "change" (not
A
TIME-change) not covered in general theory? By the way so far I see no Ray
compactificatopn. How about an easy example?
p. 5: suddenly you'say "stable",not defined. Unstable is called
instantaneous in my book. What do you do when all sltes are inst.?? How
to Ray it if you'can? That Ex. of a chain living on Q is Levy's --- look up|
a whole section in my book Folks like Henry says. as you repeated, it is
like a diffusion. I do not know why and what is "regular" diffusion?
It would help *f you can show how Ray compactifying helps in those
examples/ of Ca .tiS, bttf -ey f~v
I call "well meas." "optionally meas. see pp. 43-44 for this and
A
the Doob incidents If you can't recover his proof please ask him for a
published ref. "predictable" is adopted in my book. So "predictably means "
Do you need strong Feller property? I would like to wean4
h-vea a% icSi4.it.j ,.Pe'aatmy article "Doubly Feller .."
SrryI i- in the Seminar on St. Proc. zAfter killing
cS 4 es. ou saa domain a Doubly Feller proc re-
~J pit S mains so --- n standing p\eblem ...
._ .EGt? Can you employ it somehow.-. 0ss1- r-c




Lk 3N hit,


March 28, 2003





URGENT: Prof. John Walsh March 28, 2003
I sent you yesterday my cOmments on ydur Examples with chains. fX
Here are a few more page-by-page marginal notes/ 1 hkat sop nu .
p. 2, before Kol equs say that EVEN the qij's requiz good work by
Kol. \of course Ref. to my book. Now tell why we need those stupid PDE?
p. 3: Ex. 1. when the path reaches oD. I WOULD remind the student
that cb is NOT a Iumber hence out of'space. Refet then to your teacher's
ghostlike return from there yonder: of course in my oo k A whole section
is devoted to the minimal chain there. Unfortunately en bouna points
etc. are concerned. Ref. to my other book is required. A few specific
page or section references should be given. Ef sya% reJl c't IPC4,asC e .
If all qi=-qii are finite, we have right continuity (proved in the
2nd book actually first noted by Pittenger). I forget what we can say
abour the rest of Hunt hyp.? Maybe you should say .3 p. 4: Are you
going to tell how boundary POINTS can be defined ( Ray et al.) How about
nAnsticky bdy pt and explain why a sticky bdy pt does not branch? Hope
you will spare the pain of Martin bdy?? Even that ts mentioned in my first
book,(I think. I ft aJoyl b&raf- f' ~
Pa. cJ p. 4,lower paragraph: I don't know if Yt f(Xt) is Hunt when Xt
is: surely we need some conditio f? This is the.kind of "change" (not
TIME-change) not covered in general theory? By the way so far I see no Ra;
compactificatopn. H w about an easy example?
p. 5' suddenly you say "stable",not defined. Unstable is called
instantaneous in my book. What do you do when all sates are inst.?? How
to Ray it if you can? That Ex. of a chain living on Q is Levy's --- look i
a whole section in my book Folks like Henry says, as you repeated, it it
like a diffusion. I do not know why and what is "regular" diffusion?
It would help *f you can show how Ray compactifying helps in those
examples/ &f ChainS. a er +rn otP
I call "well meas." "optionally means. see pp. 43-44 for this and
the Doob incidents If you can't recover his proof please ask him for a
published ref. "Dredictabl." in adnon iAn 4m" "bk -- --" ----
.is adopted in my book; so
LPred ableameaske^a ]l ub P e ler Process" in Sem. on St Proc.
Please take a look at my "Doubly Fee process in Sem. on St Proc.
1985. A Hunt proc can be constructed from a Feller semigroup. Do you
do such things? It is proved there that a doubly Feller proc. killed
outside a domain remains so. Can you employ this somewhere? 1ead the
little history discreetly mentioned there. It would make me feel good
if it serves in your Parn II (forget about the functional).




( & ) 2 7/





Prof. John B. Walsh

P/elt e e(l er fr
d y~y


March 29, 2003


In the good old days departmental offices are open in the morning
of Sadday. Do not expect this for Vancouver but this is URGENT: Part
II MUST refer to Part I fir the material (faites) used!
Sometime ago I wondered about Stone-W. I was so forgetful. Of
course I used it plenty, e. g. in Sec. 2.2. [Many trivial misprints like
the E in the wrong font.] Of course I proved in meticulous details that
Sa Feller process is a Hunt process. I almost never "construct" but
SWa there I had to use separability and apparently resolvents. You should
pr ov :
prve really take advantage of this detour to implant the Ray doings etc.
q/1 This is what I called RELATIONSHIP. It would seem that you have not
troubled to sc the contents of Part I to make those RELATIONS. If
that be true the new book would be separated union. Pac b nV
I mentioned Art because I still remember a little surprise when
A he told me. In my book, a long derivation was wri en out. in Frenchy
_UyT4_. style by the dead Meyer: if all states are stable, then the paths are
a. s. right continuous and have left limitff. Bravo' Of course I did
it in my Strasbourg class and it probably gave Andrd a shock too who
was used to swipe in Luzin-... spaces. 7 of courseoo is added.
2s ,t 14&t&L ?
This is actually easy when we know that each stable state is taken in
[C,Os and there are only a finite"of theft in the finite ---Levy's great
"obvious" observation (he could not prove it; I did) that Doob totally
missed in his efforts around 1940. I should really add a cool historic
note to this (I have probably done so elsewhere). A suberb case of
intuitiontvis-a-vis "non-intuition" (for lack of a better word, not e.
g. "reason" or ,worse "logic"). Of course I did NGT bother to look at
1 your handwaving in your fax about this result. Tell me/ 9 wYn
Regarding my concern with PDE, it depends whether they play
?,A any open rAle in Part II. Only Laplace's ~howsup in my part. Did I
OLot ~d) say in one of my Notesthat Andre (N., not Paul) probably as a young
assistant felt obliged to show his Academy bosses (Petrowski came to
my mind~' read his easy book, but there are many others in Moscow)
that "the stuff" he did (c. 1930) had something to do with th fkain
Stream. So he wrote down those two PDE but then of course saw that
he could noway solve them, even INcompletely. Seed his "Ueber Analy-
tishen Methoden" and p. 26 of my Random Time book sold you for a
poubd of bacon. l ,-- S


















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URGENT: Prof. John B. Walsh


On p. 2 of my LectUres, middle of page, I dropped the word
"adapted". I forgot.
Now that you give examples with chains, it may be worth telling
your reader that even though that process may not be a Hunt process-
in fact I allow instantaneous states (maybe all are so), a STRONG MARK
PROPERTY holds in the obviously best possible way. Namely whenever
the optional place X(T) is in the original countable discrete space
(no compactification). In fact a more specific form of i is proved
in my MC chain book which you may want to take a look. I need
that specific form involving the optional timt T for many applicationE
Unfortunately I have forgotten where and why. I-&c-sw In the
ae"R"6le 2i 2
more case you heea deatJgg with/such things Sa needed.
Do you have a similar situation where the SMP holds where
it makes sense I mean if 6 is added to compactify the space,
I do not define P(6,.) and of course the usual form makes no
sense.
An Index for ypur part will be necessary)including all the
symbols ypu use (which may not mean the same as mine).
By the way, I still remember Meyer's saying that progressive
measurability is not important, the more refined optionally/predic-
tably are important. But in my exposition we begin with a prog.
meas. proc. in order to refine it to be the others. Do you do
the same? Maybe I did not understand what he meant? Do you?


4/7/2003






URGENT: Prof. John B. Walsh


I forgpt I dropped the name "adapted" early on on p. 2, middle of
page. You should compile an Index for your part including all the
symbols used. If they mean different things from my part, better
say so. )To
It is a good idea to include those examples from Markov chains.
After all this is the Horse's Head for us all oldsters. We stuA
with Sir Issac conic sections, polyhedrons before the followers
make up Luzin et al things. Maybe your latest stuff can also
show a new way of doing the Strong *rkov Property for a general
chain? In 1956 I proveahis property obtains (fancy English') at
any optional T when-andwhere X(T) is in the original countable *
space without co. Quite an achievement that the Russians aX~i*
Yushkewitz did not do )Dynkin
(they left large holes in their effort whiCh
is also published). That was just the time %ta Ray was doing his
stuff --- he was at Cornell when I gave my talk on thl' stbject(SN/p)
and took me home to have dinnerI think afterward. So let us (me)
see how you get this particular case from your new presentation.
Don't worry: just show as much as you can. But here is a question
that had bothered me in the pastwhich you may answer (for me if
not for the reader)" Take a look at my MC book, 2nd ed, p. 179
[if your library has only 1st ed. that is no reason not to find it
there on another page] I went out of my way to say that, around
line 15, the theorem with several corollaries cover more than the
usual SMP: apparently this has to do with my r.(s,t)Kthere [forget
about A ). I would be obliged if you} tell me what it is
7-A
all about, and whether similar results are pertinent to the general
Markov process? For example, are the three Corollaries on p.
17t automatically true? Not. *olt -,W;/, ( I /u


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6,,,zc


April 8, 2003






URGENT: Prof. John B. Walsh


4/9/2003
Please tell me why in the definition of FT for any
function T from Omega to [0 oo) we need to put Po+ into it?
If we use (7) on p. 16. with t=0. we get
(0 On {O=T) we get nothing. Maybe we do need F0 in PT, but why F0+?
I remember vaguely this "problem". What is it?
By the way I assume you need FT_ in your part? I have begun
my corrections. May have to skip some scribbled on my copy twenty
years ago, no longer legible or understandable, tant pis.



/









6o7

6c4~^ b*k 6 ^ ^







( April 15, 2003

John: I am shocked that you say my original formula on p. 25
turns out to be OK. The correction of b for a was made probably after
somebody in my class told me, but I must have decked it then. Now I am
incapable of doing so (having tried for several days): would you be
so good to show me how to get it from Doob's inequality I printed
for the Upcrossing of a SUBmartingale? I am really curious. As you
suggested I even looked at the unique copy of my 1st ed. of Course:
there are several forms of such crossing number estimates but none
of them can be easily converted to get what I used (foolishly) on
p. 25. Please show me. I shall rewa'fd you by giving you a copy of the
3rd ed. incl. measure and integral.
I am around p.80 now and found another correction in my copy
that I can no longer check. I'll try it once more later and may have
to either ignore it entirely or ask you to take a look --- itis
east stuff.
Do you have a good grad student whom I can hire to check cer-
tain places? I used to have but only one or two were reliable
Now of course none. By the way I got a letter from Doob (about
Meyer --- did you go to his memorial ?) He is 93.









___. ck^r c Q"c )





,82 z z 4C)74





Urgent to Prof. Walsh


John: Please take a look at Prop. 8 on p. 84, and see rgpp. 5
on p. 82 for f. I made some correction there but do not see it now.
Can you see) Please reply --- I am about to finish my parts and
read what you sent me recently about the chains. One thing: except
for stickiness and such fancy doing, my old M. C. book (2nd ed.)
has all the examples you discuss there and may be a better reference
for the average reader. I already told you the example of all qi=o
is not Levy's (for once his intuiJt4tion failed, as I reported in
my book for one (more?) pound of Canadian bacon. See 1il. cit. oov,


AyqA24 42


6"),2


(DO;jL
Lojt


P. Ks)


April 26, 2003
< ^




4/29/2003 12:05 PM FROM: Fax TO: 101594516508570208 PAGE: 001 OF 001




Hi, Kai Lai,

Propositions 5 and 8 look ok to me. I suppose you could mention in Prop. 8 that (10)
implies that f is superaveraging. Can you be more specific: what in particular should I be
looking for?

Cheers,
John


Sorry to trouble you. On my teaching copy I scribbled under
(15) on p. 85 (not 84 on my last fax) and added things on the last
line
f = liml lim ... J
k n
Can you see if it is OK?? I really can't read it but will leave
it alone with your support, or make changes if necessary. It is
too late to "improve" what is printed there, I just want to make
sure it is not wrong. Please say.


Good news for your discussion of Markov chains, see p.56, No.
3 with page reference.

Forgot if I asked you if you need all those universally measur-
able stuff? Where? I remember that came from Meyer: Hunt only
introduced "nearly Borel". But in your generalization what do
you need?



KdcA ^








D to John B. Walsh May 2, 2003

John: I caught it, tant pis. Did you realize that on p.85 where

I asked you to check t>t I used the fact that min(fk) is exc.? Of
(see my GOOD Index)
course I proved it, but only much later and I went out of my way to say

that it is a hard nut. Maybe you know an easier proof??? So my 20-year

old correction will stand: use fk instead. Unless you know a faster

way. By the way I have''braching point on p. 9. Maybe you should use
A
P0 in your example of such a point.




Have you seen my (unique) paper with Glover on left continuous

MP (refereed by Andrd). It would be nice if you can use any of the

results there as I recall some are rather tricky. Some time ago I ask-

ed him if he had any application. -je said no, but you knww he is

now some kind of "dean" --- Art was provost.







Prof. Walsh


Please look at p.221 top. I scribbled there:
It follows that the max pr. holds without requiring mu to
have compact support by considering mulK (mu restricted
to K??)
Does it mean anything?
I will send the corrected galley some 250 pages to you, but
please make a revised ( by you) copy and mail one copy to
me, keep another with you to be mailed together with your
MS to Ina. Is it OK? Surely your good secretarial support
can do the machine-work for you. I can't do it myself.

The French potential theory sans P has something like
supermedian if not excessive. They do not have right-con-
tinuous paths. So how do they prove that the min of two
of their super... functions are super.. too? Is this true
for positive superharmonic functions???











22. Z6<+









Vancouver, 5/5/03
Hi, Kai Lai,

Here is a section on stochastic integrals and local time, which I'll send you while I'm
slaving away on the duality stuff. (Either I'm missing a few pages of those Paris lecture
notes, or I finished the lectures before I finished the subject, leaving a few questions
dangling.) And of course I keep thinking of more things which should go there. Anyway I
have to figure how to prove a couple of extra things which are no longer as clear to me as
they once were. In particular, why did I think it was completely obvious that "semi-polar
implies polar" was equivalent to "fine topology equals cofine topology except on a small
set"?

Anyway, I doubt that the enclosed section on stochastic integrals and local time will
make it into the book, unless you can think of some use for local time. If you have any
ideas, let me know. There are lots of uses, but not a lot of time, so we need something
short and sweet, but still important. I suupose we could always use it to construct those
examples of Markov chains with one or all states unstable, of course, but that wouldn't fit
in so well with the rest of the chapter, and it's not enough by itself to justify all that work.

But it's mainly for your interest. You said I could have five pages to develop stochastic
integrals. That was enough for the L2 case, but by the time I got Ito's formula for local
time, it took eight. I'm bored always repeating the usual construction, so I gave a different
one, constructing the integral directly as a martingale. The main existence theorem is
Theorem 1.5, which states that a couple of Hilbert spaces are isometric. It simultaneously
gives the existence of integrals, the L2 bounds, and Ito's representation theorem, although
I didn't prove the latter. Not good pedagogy, I'm sure, but I think it's cute.

Maybe I'll send a zeroth draft of the hpath/duality chapters to you shortly, incomplete
as they are. If nothing else, it ought to put your mind at ease about the connection of
this stuff with duality, since I have a nice story about Markov processes in duality and
electrons and holes in semiconductors. Damned if the story doesn't even make a couple
of the balayage formulas look more reasonable. (e.g. why the equilibrium measure of a
set sits on the union of the set and the points co-regular-not regular- for the set.) You
can't see that in your case, by the way, since you assume that u(x, y) is symmetric in x
and y, so that the process is self-dual and co-regular equals regular.
Cheers,







h^ c^+^H^ k [2t a t h
o aOfc a







s5.2. SMPta r dr inml y .4 221
We am aow ready to etal anotder meaor priaci kinow a the
Maaris Max himx PriJkpfk en flow. For y -iteW mIeasre p
supported bytfc iK, we hae
(M)
4 pA(x) sop Ux)(M)
ThI prairie a be ittivdy obvious to a phyiS, mir it ys that the
pommel inu led by a dYarp.is gmea whI th dawp li Yet is poo
s mas to dpm a m eort otdof dlity m pria a the diribaM of
the -ra 'Thib to bow that plyrical proin w rr an ihert
dualiy ll d byw the alfdmtim
Thmm The mifiM pk*f(f (M) AMb rider (S (T,) m (UA
Prf. 'T hi mhiag to dm ifte rid i mbr of((M) is ilaht so w
my ppmu it fit mid eql to M. For g >0 dd~a t ie
B- {xe BUp(Ax)s M +s). (12)
Sinm Up is ewi udhr (UI), it is abl coetiodeom (Corolay 1 to
Theirom I ofp.5) amd so B is aly dokd. The e ooaliaity of Up aso
7. ii that K c lr (whyr) Thifr wre wive by Theors lad Fobimi:
k Cev-9 PUp(x) fPas(x, y)M fP.(y, xO(dy)
'(y .,x <,, ) Up(- x) (13)
3 beMae l c K d f dbr ye a K, P(y,.x) (- a(Y .4tivit ymn e .
S7O PA(x,-) hbas nppo in I by TlhBor 2 dA .4, aMe
*i adly domd. Hmes we have
P /Upx) Pa(xdy)Up(y) up Up(j) ; M + 114)
Putting (13) ad (14) t-eher we concede that Up M ies m i aurbitsry.
[*
e arpmaet ledimng to (13) is piiem and maoedd bow.

CaSiey. For ny #-in* mIrmee p aawk da L c r, we tah
U PPUp. (IS)



_ _. ,. .f ,1 ,






fftea- Aoapl6L
fMe ^o -Lo ^ t4~ Re I Su>

"Iel 'e de7CCa3er^,c~ C

SU".L F!s








Prof. John B. Walsh
May 11, 2003


Sent you a copy of the page where (M) is proved, with
marginal notes. As far as I can read it now, after over 20
years, I do not know why it was assumed that the measure mu
has compact support. Can you duszover where it might have been
needed?? I have now looked over five books including Brelot,
Rao, Landkov, ... Nowhere except in a first discussion in Brelot
where he began with a bounded open set (I do not know why) do
any of them specify a compact support. In case you ahve any
such book, such as Doob's which I did not check, you may try
to find any hint for my past doing. Anyway if you can guaran-
tee that the proof given in my text goes through for any sup-
port of mu, I will just delete that stupid reduddant condi-
tion there and leave all the rest alone. NO SWEAT.
Just in case I really needed that stupid condition for
the given proof there, I will have to add a few lines to get
the general case (more or less as you showed). That woyld
be pain in the neck. Now I wonder how many other such
stupiditoes







Prof. Walsh May ll, 2003


John: My last fax may have confused you. I read my
proof on that page and did not see ee*" WHERE I used the
compactness of the support of mu. If you agree please
fax and I'll just delete that stupid unwanted assumption.
No Sweat. If you do find that I need compact K there I must
add a few lines as you showed and state the result in the
general case. Worse luck.




May 13:
I am waiting for your reply Jo the above to finish my part.
You will have the corrected galley before you finish yours,
and I expect various references (if need be) to s part
if you can insert them somehow in the t~y part. But that
would not be absolutely necessary: The way out is for us
to say in a new Preface (abs. nec. this!) ~csy something
about the connection-and-lack-thereof so that the good read-
er is.at least warned.
I am Mailing you the first six pages of the stuff on
.chains with some marginal notes. Nothing substantive and

some may be out-of-date.
I have only glanced at the Ito stuff you sent.
One way out is to put it in an Appendix! Nowadays it
is so popular with the financiers that any dropping of
the name may sell a few more copies (to rich brokers).
One historical note you may not know: Doob told me it
was he OW) who put the martingale connection in the
integral. I was IR W ln proof-read Ito's original
memoir in the AMS and know that no martingale made its
appearance. It is tries probable that. had the idea
without spelling it out. So what you wrote there may
re ~re a footnote of "historical perspective". Read
also in my book with Ruth my presentation of McKean's
way of doing something with max Bt but I forgetAII-t
ab oJ local 6:me, (?)


























This graph, originally from the Washington Post purports to compare the income of doctors
to other professionals from 1939-1976. It surely conveys the impression that doctors
incomes increased about linearly, with some slowing down in the later years. But, the years
have large gaps at the beginning, and go to yearly values at the end.





5/15/2003 10:38 AM FROM: Fax TO: 101594516508570208 PAGE: 001 OF 002


Vancouver, May 14
Hi, Kai Lai,
This is the second day of trying to get this to you. Your machine seems to be sulking,
and refuses to talk to mine. And the fax is getting longer hourly, as late-breaking news
arrives. (The last paragraphs have changed three times already as I find counterexamples
to the previous statements. Tomorrow... Who knows?
Classes are over here and my exams are finished, and I'm not getting into the depart-
ment every day. So I may take a few days to reply to your faxes.
I've been working on the final chapter of my part-potential theory under duality-
and it turns out I'd left my Paris lecture notes unfinished, so I've had to do some hard
work to show that under the proper (strong!) conditions, for an excessive function h, that
(1) the sup of potentials dominated by h on A and having compact support in A
(2) the inf of excessive functions which dominate h on a neighborhood of A
and
(3) PAh
are all the same. And when they are different. It all has to do with hypothesis (B) and
quasi-left continuity. This is admittedly dotting the i's and crossing the t's, but I'm feeling
pretty good about it. The annoying thing is that I have no idea if all or any of this is new,
but as the key tool is the exterior reduite, it might be. (I don't even know if I invented the
exterior reduite, but with that name, probably not. Do you know? Have you ever seen it
elsewhere?) You'll be glad to know that I needed Choquet capacities and could duck half
the proof by referring to Prop. 3 and Lemma 4 in 3.3.
By the way, this chapter is overlapping your chapter 5, so I said that for comparison,
we would do the same thing by more classical methods there.
Next in line is the "semipolar -= polar" problem. As you say, it is a symmetry
question. Basically, it is equivalent to "fine topology = cofine topology." That's true in
your setting, of course, since you have a symmetric potential density. I have to work out a
few details about exceptional sets to get things to look pretty, but it should come around
shortly. I hope.
It's fascinating that Doob put the martingales into Ito integrals! How did Ito do it
without them? He must have proved a special case of the martingale maximal theorem to
get the convergence of the integrals of simple functions.
I'm cool with any plan to make use of local time, etc., but I wouldn't expect to fool
even rich brokers into wading thru chapters on Hunt, Ray, and dual processes in order to
get to the definition of local time! McKean does have a nice stochastic integral derivation
of local time in his book-substantially what I gave you-and he has a clever way of
constructing the stochastic integral itself, using an exponential maximal inequality. That's
probably what you asked about. But in the end I don't like his method as well as Ito's.
The questions you asked in the May 7th fax require a trip to the library, unless by
happenstance I have the answer in a book at home. It'll probably take a few days, and
in fact, my plan is to try to finish my notes, send them out so that someone will have a
chance to tell me about some of the most obvious errors and misprints, and maybe help
with the attributions. (Mostly missing at the minute because, while I know the math, I
no longer remember where I learned it.) After that's done, I'll need to spend some time in








5/15/2003 10:33 AM FROM: Fax TO: 101594516508570208 PAGE: 002 OF 002


the library looking things up, and I can get to your historical questions then. Keep asking,
but don't be surprised if I don't answer. Maybe you should make a list, and give it to me
when that time comes.
For example, I don't remember what Brelot's hypotheses were, but if he assumed that
his cone consisted of lower-semi-continuous functions, that would do it. Or did he simply
assume the cone was closed under mins?
Doob states the maximal principle for Borel sets, not compacts. (Pg. 67 of his book,
for Brownian motion.) His statement is more general, and he calls it the Domination
Principle: you want part (a) with v = constant. I imagine that the reason that compacts
usually enter the statement is that in axiomatic potential theory, one often wants to use it
as a hypothesis, not a conclusion-"Let C be a cone of potentials satisfying the complete
maximum principle"-and it is enough that it hold for compacts. And of course, if the set
is not compact, you have to define what you mean by support.
I think your proof is OK with a trivial modification: you have to replace supp(p) C K
by "u sits on K" or "~(Kc) = 0" since if K is not compact, it can carry p without
containing its (closed) support. The only place I can see that compactness might enter is
your conclusion that K C B', but that's OK for any K, as long as it contains no branching
points... (chuckle, chuckle)
I just noticed that by taking h = 1 the maximum principle drops out of a theorem
I just proved (namely that (1), (2) and (3) are the same if you have Hyp (B), quasi-left
continuity AND if h is a regular excessive function.) I think that the complete maximum
principle (having it hold for all bounded h, or at least all potentials) requires "semi-polar
= polar" but we'll see. (The condition is sufficient, since it implies that all excessive
functions are regular. I just don't know if it's necessary.)
By the way, in the set-up I have-oh, well, the set-up needs to be slightly strengthened--
quasileft continuity plus hypothesis (B) equals "the branching points are co-polar and the
co-branching points are polar." See how nice branching points can be?

Cheers,

John






9~Qr am Prof. Walsh
Friday 5/15. 2003

John: Your reply is not Clear to me. On p. 221, line
3, I deleted "compact" so that K is any support of mu. This means
Mu(KC)=O. Some people (not Doob in his 1984 book, where wrote
"supports" but FORGOT to say what they are.) like Dellacherir"Meyer
define "THE support" to be the complement of Kzix~~t xi~f xkiXl un-
mxxxamztyxmayxixfxamxxEHixxxKi the union of all open sets U such
that mu(U) = 0. Is this your definition, if not what is it? Now
on p. 221, after (13), I wrote "because j1 CK", hence that support
\& is a subset of Br (see the line above (13) where I wrote
"why" --- because I think of right continuity or if you will fine
continuity. Hence any path starting from that particular A will
hit B at once and so by definition of B in (12), 64O. as in text.
1 asa Ra 7 do not see why my original i LC is wrong
although I can change it to "'K supports L ". Please explain.
There are many different "supports" of mu, it seems to me the
one defined by Del-Mey in their big bookT(pkease check) is the
smmalest of all supports. Is this true? If we denote tha tky
\ .taienfty (arbitrary) support K must satisfy LFQ as I
wrote, and the supremun of U,.) over 11 must be < the supremum
over K as written on the righthand side of (M) on p. 221. ISt
M0Bie8 I am really confused. )ypCl(i by jS




f-P 4







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JPZq~)S I~







Prof. John Walsh: Nicht Urgent


May 28, 2003

I got your fax about h-transform etc. I do not recall
any connexion between this and reversing. In my book with Zhaotal-
ready in second printing/edition! there is a lot of h-BM there!
In a Lip domain and using the Martin kernel K(.,z) as h. we prove
that the transform is nothing but conditioning on the first exit PLACE
X(T). If you can generalize this it may even be new. But I quit.
aCt The rest is for Yuke.



Dear Yuke,
Glad to learn that Kevin (whom I remember seeing when ar-
ound three of age) WRITES and has been to Guatemala. We were there
some twenty or twenty five years ago and I like it very much except
the Guatemala City. Antigua is fabulous and ifs GENUINE coffee is
one of the two or three best in the world, not available here (all
fake or else). Atitilan (maybe there is another t somewhere) was
praised by Al4pus Huxley as "even" more beautiful than Como. I do not
agree but that may be due to the fact that I (we) have been to Como
Some ten times and to Atitilan only once in a tour. See wha Kevin
WEITES wav put my comment versus Alious in there. I am hoping to
go to Guatemala again .... Best regards to you and your sons.








Wl cPro Walsh June 5, 2003

I spent 15 minutes looking for your definition of "minimal
excessive function" and did not find it. I bet there is none. Of
course I do not need it.
One of the stupidest custom of typing/printing is to omit
the page numbers on certain "beginning" sheets --- I never under-
stand or guess the PRACTICAL reason for this: maybe your good expe-
rienced st~hographer can tell you? By the way, is "stenographer"
in vogue in good old British territory? 6?* .
Shizuo gave me most of his Imperial College shitty-paperg
reprints containing his doings for the Nevaliva theory (??) that
becomes a treatment of the boundary value problem (your teacher was
a fast followers c. 1944 --- before the bomb). Not only I read
it but I reprinted a whole page in my renowned Green, Brown,
[NOTABEBE: do not let your good stenographer omit, by stupid cus-
toms the important comma after Brown' Some 93 percent of my
correspondents omitted it probably never saw it.] and Probabili-
ty, second Edition, Mb w come I never saw any "holes" in your
scanning? Take another look and point out to me where the holes
are??? Did your teacher, not to mention Andr4 or Gil, see those
holes?
I found an unmailet envelope contain the pages on chains with
a few unim rtant marginal notes by me. When I was told by
the post office (its 800 number telephone) that Canada costs
85V (China or France only 80/) it was not mailed because I do
not have 85/ stamp. I will ask Isolde mail it sometime.
You (We) are contracted to turn in the MS in June. Just
phone Ida. She seems to like you a lot.






(66+ 23-




TRANSMISSION VERIFICATION REPORT


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.06/04/2003 14:05


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Prof. Walsh
June 8, 2003

John:
Please show am the proof that I.-yl-1 is minimal excessive.
If you have Notes as I did, explain in engineering detail how a semi-
conductor works the holes. host readers would not know.
Call Ina about the delivery of our MS.
On second thought, it would serve the reader better if you write a
preface to the new edition, say what you want, and let me make some
suggestions if'any. This will be printed after (or before?) my old
preface (with a singQle misprint corrected as shown). The reader
may read them in either order or not. I know most American students
do not read prefaces (did you: but the "experts" may. Too bad
Andre cannot, Doob might (if you sern him a copy of your preface),
and I suspect Gil won't AFI assume he is alive but nobody told me).
You should add (tems to the Index, that is necessary service and as
you know really useful for any such book.
When the galley comes, I will proof-read my part, and scan yours.
There will be misprints and mistakes. There were too many small
misprints in the old text --- my fault but Wu Rong simply was not
up to it. If you have/ good studentkyou may employ one to help
you.
As I said the deadline may be postponed some, but "hopefully" not
too long. Againteak to Ina. Good luck!
I will mail you the few pages on chains, not important.
What result by Pittenger you mentioned?





Cci
C~oi^-^e^^ 0-~




6/11/2003 6:00 PM FROM: Fax TO: 101594516508570208 PAGE: 001 OF 003


Vancouver,June 6, 2003
Hi, Kai Lai,

By coincidence, I went into the department today. Don't come to expect rapid answers,
it'll probably never happen again.

No post-dating-you got it shortly after I sent it. I'm a night-owl. The thought of
getting up at 3 or 4 am is frightening, but I have no trouble staying up that late.

Great that you found the reference to the MMP-you really have to know what you're
talking about to recognize it, but it's there, and the invariant field is just the right one.
(Damn, I should have called attention to that in a couple places in my part, since I did
introduce the invariant field, and didn't make much use of it afterwards.) Yes, by all means
write something about it.

Old friends first: I don't know Dellacherie's address or phone, but can probably find
it. I can't remember complaining that Meyer preferred Claude to his other students; it is
clear now (and probably was then) that he was the best of them all. I'd probably have
been talking about Maisonneuve, who was-is-a good friend.
The inventor of the predictable sigma field (and perhaps the predictable stopping
time) was Catherine Dol6ans, now Catherine Dol6ans-Dade. Andr6 had divided them into
accessible and totally inaccessible times. Andr6 was always very scrupulous to credit her.
(By the way, Doob complained bitterly about the French stealing Illinois' best algebraist
when she married Everett Dade and moved to Strasbourg. I didn't ask Andr6 about the
American's stealing French probabilists when they moved back to Illinois.)

About the first-name last-name question with Meyer: don't forget that I arrived there
the year after Les Jours de Mai. The student-teacher relation had changed entirely, the
profs showed up in jeans instead of a 3-piece suit, and the grad students tu-toi'd them as
a matter of course. I think you were there slightly earlier, when the rule was as "vous."
As a matter of fact, I never did get comfortable calling anyone "vous" in French, a great
source of embarrassment to Joke at least once, when I addressed a venerable 80-year old
matriarch as "tu."

I got a reply from Ina. A bit of a problem: I'd asked how much space I had. She
replied that she hoped I could do it in 110 pages. I replied 1 was up to 156 and counting.
I just got a reply to that one, saying she hoped I could do the remaining notes in less than
13 pages. (That was a great solace, I was afraid I'd have to drop my section "Death and
Transfiguration: a Fireside Chat.")

That may not be a problem: I find I have a great desire to keep it short and finish
with this. But it also means that I won't do anything on Brownian motion, a pity since
after all that work, I should apply it to something concrete. And I'll have to avoid brilliant
second thoughts which just have to be included.




6/11/2003 6:00 PM FROM: Fax TO: 101594516508570208 PAGE: 002 OF 003


Forward to the Past!

(1) About the MMP: I can't remember the aha! moment, but I am sure we invented
it. (And so we're responsible for the name, too. No regrets permitted. Actually, I like
it-its a dig at mathematical names: the strong this, the weak that...always the extremes,
why not the moderate something? But then, I suspect it was my coinage, so I have to like
it.) Anyway, I certainly didn't learn it from the literature. If you think about it, there's
little cause for it-certainly Hunt didn't have a chance, since with quasi-left continuity, the
process is continuous at any predictable time, so the moderate Markov property is part
of the strong Markov property. About the only place you'd be likely to run into it would
be in Markov Chains, where the first infinity is predictable and the SMP certainly doesn't
hold there. But in that case, you don't need a general theory to deal with it. So I think
we're pretty safe there.

(2) Unilateral limits: my take on this is, when in doubt, make a Ray-Knight compact-
ification. (The Ray process IS the most general Markov process.) Then the left and right
limits will always exist. In the Ray topology, the left continuous process is always moder-
ately Markov. (Which means, by your definition, that a Ray process is both strongly and
moderately Markov.) Under duality, both processes are moderately Markov. (Thm 5.6.1.
I suspect that's new, tho it certainly shouldn't be. But I haven't followed the literature,
so I don't think I'll make any claims. Actually, I have the same problem with a number of
results.)

Now I have some historical questions-and I'll probably have a lot in the near future,
as I try to do the notes on my section. (If I don't bail out entirely.)

(a) History of reversal. Kolmogorov, of course, but his reverse wasn't homogeneous.
The first real reversal from a random time that I know of is Hunt in 1960, with his
approximate Markov chains, reversed at last exit times. This is surely where Nagasawa,
Kunita, Watanabe, et al. got their ideas. It certainly influenced me. However, there might
well have been some reversal from a first-hitting time before-did you do it for Markov
chains from a first infinity, or something like that? This reversal stuff had been floating
around for a long time, but when did it first get written down, and who did it?

(b) What about the double-reversal arguments? Did you ever use it to discover a
theorem, prior to our article, natch? Lots of people did, I think; it was one of those things
that were obviously true, which you couldn't use in a published work since nobody had
made it rigorous.

(c) I proved a bunch of theorems about Markov chain paths in what I sent you, and
I know some of them go back to specific individuals. I remember you mentioning Donald
Austin, and I know that one of them goes back to Art Pittenger. Do you think you could
go thru that section and tell me. (Yes, I know that I can get it from your book, and I
will if I have to, but it's be easier for me if you'd point out any of the results which were




6/11/2)03 6:00 PM FROM: Fax TO: 101594516508570208 PAGE: 003 OF 003


reckoned to be brilliant work at the time.)

(d) Whn you're tracing the MMP, don't forget that last-exit decompositions are a
form of Markov property. Might be worth mentioning. (Hey, its a first-hitting decomposi-
tion for the reverse. At the moment, I'm only devoting one sentence to it, but I wouldn't
mind saying more.)

Cheers,
John







Prof. Walsh


June 12, 2003

Did you see the footnote on p.44 of my Random Time?
I did not say-so if'our paper to -^iel Andre. L'etht:fGCest mo):-proved
a reverse Markov process has stationary transition --- Meyer
failed even to do this in his (later withdrawn but I have a
copy) Seminaire cited in the footnote. Read it. My reverse
is from the first infinity of a chain, and that is the model
in our article. I need some smoothness of the az i
e- which you supplied witr triple exponential prolonging.
I did not vaunt these matters in my Notes because
MC was only mentioned in simple examples (Paul Levy chided
me in his review of my Course for not treating M,C ).
History is a tricky thing --- better keep mum when you are
not sure. Assuming you have a copy of the Proc. ICM at Nice
(where I ate boursin f@r the first time, grace a Yoke [Yuke?
maybe Joke in Dutch: ask her please), the history of MMP is
told in style and my prediction regarding its prevalence
now seems accurate. In 1969 it fd called "moderately
strong", the precise name may welmIeade its appearance in
my talk at Nice.. "f you want a reprint I can send one.
Regarding double reversing of course I did it some-
where but now forget where. Be careful: one needs a
primary condition something like the reversal time be
finite or somethinge In a conversation with that Jap
in Zurich I forgot to say so and he, japlikes jumped
on it.
You did not understand what I recalled about Meyer +
Claude: your complaint was that Meyer preferred C's
counterexample to yours. I found that error in C's book.
Repeat: for any remembrance of the past, DON'T say
it unless you can find the record in print. Yushkewitz
told me a story about "Eugene claiming some result not
his, with kudos from his coterie, but the true story
is in print. /If you contact Art Pw, let me know.
My regrence on p. 74 of Lectures is faulty:
that book was published before 1969 and does not
contain the name MMP, only the result written by /
Meyer. Hope this ends our chat about history.






,knee./X ;Z 33I-
Additioi'be Bibliography
K. L. Chung
~l] Probability method in potential theory. Proc. Conf. Poten-
tial Theory (1987), Lecture Notes in Math. 1344, Springer
Berlin.
[12] Green, Brown, and Probability Brownian Motion on the
Lines World Scientific Publishing Co. Singapore 2002.


J. L. Doob
[ Classical Potential Theory and Its Probabilistic Counter-
part. Springer 1984

r91 On the boundary theory for Markov chains, Acta Math., 110
(1963),p.19-77; 115,(1966, p. 111-163.
[10) Bpoundary behavior of Markov chains and its contributions
to general processes. Proc. International Congress of Mathe-
maticians, 1970~ p. 449-505 (with one picture). published
by Gauthier-Villards 1971.


John Walsh:


June 13, 2003
13 and Friday


Please make sure the above additional Bibligrpahy is print-
red, at an appropriate place. Other Bibliograohy needed for your
part may be added or (easier?) printed where your part ends? Up
/ to you but tell Ina our choice. The format has been changed as I
v d added "p." wkich was omitted in the old edition. Now please
figure out an optimal mode to insert the following additional
Note (as we discussed in faxes),an Vpper. of )(P ,-k be e'th fe
Notabene: the nane "Moderate Markov Property" made
its first appearance in Chung [10] as far as we can
I trace it. Atkkxgk Indeed it was fortuitously observed
_____6 during uwxx page-proof maxxXKXXza~i of Chung 19]3 pp35, and
the lefthand approach is liA&o re for crossing the barrie
' V ~bret eri Later when Walsh saw its iR~ RRR*x natural ------
4k- rlME wIwhen time is reversed (right becoming left) in Chung and Walsh
r]de iag ardOje ilml 4stnuA rwa

name "Moderate Markov Property" was introduced. Since it
is much in evidence in thi-- new- di-ten aet this story be-
addcL to-d here, )A nevi chpers is
'^*4vr4f. pleasure ai- 4ellv )1"Ti;t,

















JUw~o t an /D P;Z 3' S- f

K. L. AdditionSt Bibliography
K. L. Chung
01] Probability method in potential theory. Proc. Conf. Poten-
tial Theory (1987), Lecture Notes in Math. 1344. Springer
Berlin.
[121 Green. Brown and Probability Brownian Motion on the
Line, World Scientific Publishing Co. Singapore 2002.

J. L. Doob
4 Classical Potential Theory and Its Probabilistic Counter-
part. Springer 1984

F91 On tLe boundary theory for Markov chains, Acta Math.. 110
(196*),p.19-77; 115.(1966, p. 111-163.
[101 Bpoundary behavior of Markov chains and its contributions
to general processes, Proc. International Congress of Mathe-
maticians, 1970. p. 449-505 (with one picture). published
by Gauthier-Villarde 1971.




















Additionirt-~ Bibliography
K. L. Chung
-1] Probability method in potential theory. Proc. Conf. Poten-
tial Theory (1987), Lecture Notes in Math. 1344, Springer
Berlin.
[12] Green, Brown, and Probability Brownian Motion on the
Line, World Scientific Publishing Co. Singapore 2002.


J. L. Doob
[4] Classical Potential Theory and Its Probabilistic Counter-
part. Springer 1984


[9] On the boundary theory for Markov chains, Acta Math., 110
(1963),p.19-77; 115,(1966? p. 111-163.
[10] Bpoundary behavior of Markov chains and its contributions
to general processes, Proc. International Congress of Mathe-
maticians, 1970, p. 449-505 (with one picture). published
by Gauthier-Villards 1971.



















Additio~Sbt Bibliography
K. L. Chung
1)3 Probability method in potential theory. Proc. Conf. Poten-
tial Theory (1987). Lecture Notes in Math. 1344, Springer
Berlin.
1121 Green. Brown, and Probability Brownian Motion on the
Line, World Scientific Publishing Co. Singapore 2002.


J. L. Doob
1 Classical Potential Theory and Its Probabilistic Counter-
part. Springer 1984

r91 On t.e boundary theory for Markov chains, Acta Math., 110
(1963),p.19-77; 115.(19661 p. 111-163.
C101 Bpoundary behavior of Markov chains and its contributions
to general processes, Proc. International Congress of Mathe-
maticians, 197l. P. 449*505 (with one picture). published
by Gauthier-Villards 191.






/6?/SA /


1


Addit o Bibliography
Addition'** Bibliography


(2dZ


- K. L. Chung
I / Probability method in potential theory. Proc. Conf. Poten-
'1 dtial Theory (1987), Lecture Notes in Math. 1344, Springer
Berlin.
1 12] Green, Brown, and Probability Brownian Motion on the
Line, World Scientific Publishing Co. Singapore 2002.

J. L. Doob
[41 Classical Potential Theory and Its Probabilistic Counter-
part. Springer 1984

[91 On the boundary theory for Markov chains, Acta Math., 110
(1963),P.lA-77; 115.(1966? p. 111-163.
I~J( [10] Bloundary behavior of Markov chains and its contributions
S--- to general processes, Proc. International Congress of Mathe-
maticians. 197,. P. 449-505 (with one picture). published
by Gauthier-Villards 1971. /
by autie-Vilars 9V1 I


-T.Yh 4 %( witL Af ia h 6 '
^**/^^lY.- h q


A) "A '.B


Notabene. The name "Moderate Markov Property" made its
first entrance in Chung [101. It was fortuitously observed
during galley-proofing of Chun4[9;P.35],when the boundary was
crossed at the "first infinity" time. John B. Walsh saw its
natural pertinence when TIME is reversed, 4" Chung and Walsh
[l11]where it was called "moderately strong". Since this left-
handed variety of Markovian behavior is munh in evidence in
this new edition, its L ..,,:. i: i.,. ere for the
pleasure of the company .
June 15, 2003
John: This is my "best effort". Any suggestions are
welcome but please fax me soon. You write well, generally
speaking. Notabene must be printed as is: you knww what it
means in Latin.

K fl1^ e^U 4
^w^ ^--^t.


III ...... 1


6 S


tL W"j 06p7c^ ye &_ ..






Prof. Walsh
June 16, 20'3
John: Please discard the fax I sent yesterday. une
Insert this revised one on my p.235. The Notabene (must print

as written, NOT N. B..) may be inserted where you choose, but let
me know. This is my final act. Ciao.







Additiorgr*G Bibliography
K. L. Chung
- l] Probability method in potential theory. Proc. Conf. Poten-
tial Theory (1987), Lecture Notes in Math. 1344, Springer
Berlin.
[12] Green, Brown, and Probability Brownian Motion on the
Line, World Scientific Publishing Co. Singapore 2002.


J. L. Doob
14]
1 Classical Potential Theory and Its Probabilistic Counter-
part. Springer 1984

r91 On the boundary theory for Markov chains, Acta Math.. 110
(1963),p.19-77; 115,(1966! p. 111-163.
[10] B1oundary behavior of Markov chains and its contributions
to general processes, Proc. International Congress of Mathe-
maticians, 1970. p. 449-505 (with one picture). published
by Gauthier-Villares 19'1. 1



Notabene. The name "Moderate Markov Property" made its first
entrance in Chung [10]. It was fortuitously observed during galley-
proof of Chung [9; P.35], when the boundary was crossed at the-J-
"first infinity" time. Later John B. Walsh saw its natural perti-
nence when TIME is reversed; see Chung and Walsh [1; Section 6],
where it is clumsily called "moderately strong". Now that this
left-handed Markovian behavior is muchtevidence in the new edi-
tion its origin may be told e a M~MENTO.
C xS)







June 17, 2003
John:
Since the book has "Time Symmetry" in its title, and reversing seems
all over the places it is de rigueur to record some history about it. You
should write it since you wrote most of it in the book [I only casually
mentioned it.] But please send it before you send it to Ina, for a check.
No we do not have to start with Kolmogoroff for a finite state chain
which is a dual rather than reverse: read Feller and my M. C. book (my
student Derman extended it to the countable case). READ first the footnote
on p. 44 of my Random Time, to learn the status at that time, unless you
'ei. have Meyerts original articlewhich was withdrawn later That is PRE-his-
jq/q tory. Then read the first page of our paper(196a Next, my Nice talk in
the ICM proceedings. Read also pp. 271ff of my book on M. C., second edi-
tion. I had forgotten tYI note on Martin Boundary where I reversed from
the first infinity. Here is the not-yet-recorded history.
I showed you the last bit and told you that I needed a density of
the reversal time, the i(t) there which I saw also in your t4&. You
supplied it by exponential three times. The MMP of the reverse was your
great deed ... But we faxed about that already.
After our paper Meyer wrote his "commentary" with some "improve-
ments"? That is for you to check. I was very kind to him in my footnote
cited above. Yerf;n ts ?
Let me know if you need any informations [French and Italian
both have the plural] I assume you have my reprints referred above
-uch as the ICM article with a nice picture which you may wish to point
to as branching etc. TIME is running short, so I am mailing this with the
pages on M. C. you sent me months ago, but will also fax you a copy of this
if need be. Post to your labour-strong country may be slow, I had heard
our right-wing politicians say --- as well as your health care persaipel?
If you have ever been hospitalized you may know. Thank Heaven I have not
been to a hospital yet since my eye operation. Cheers.




tjei o ry







Prof. Walsh, GENT June 18 or 2O3

John: An important reminder. You should write Historical Notes for
TIME SYMMETRY viz. reverse/dual whenvyour part begins these doings, which
(W to eva
seem to me to occupy most of your space. Going back to Kolmogoroff is not
enough: ftr the finite chain there is statiinary distribution (later ex-
tended to the countable case by MY student Dermans did you know?) and so
there is "no problem". You obviously were not aware of the long article
by Meyer cited on p. 44 of my RANDOM TIME book. I am sure you have a
copy of that Seminaire because it was published when you were in Strasbourg
(Moulinr). There he clearly stated the difficulty of getting a homogene-
ous reverse (i. e. with stationary transition): check this against my
memory. The Japs cited on the first page of our article( 969), following
Hunt (discrete time) reversed from some last exit time but I do not re-
member if they proved the homogeneity of the reverse. It is for you to
check. The problem of true reversal, namely path by path, was at the time
regarded either as "obvious" or "impossible and unnecessary". In my Lec-
tures I did not reverse (except Doob's from fixed t) so I said nothing.
You should tell some and send it 82% me for proofing. Do not forget
Meyer's article on ours which has some merits, and those other papers you
wrote with Smythe et al (which I do not remember at all). There is a Russi
at Cornell (in 1994 when I was there) who apparently did something ....
Afftixxast If iTere is any literature after c. 1990, I know zero.
There was an AMS conference on time reversing at Santa Cruz, where Azema
(and his girl friend), Heinz Bauer (dead), Dynkin, Getoor, ... attended
Unfortunately I do not know if there is any record of the talks. Anyway
as usual everyone just talked about his own junk.
I will mail you the few pages you sent Ke on chains with my own
stamps but am not sure how much. In older days such mail was delivered
with additional postage required. Let me know if this one "works". I am
amazed how high the postage to nearby Vancouver.
P. S. Example of lost history: neither Doob nor Hunt reversed zxxixx
except as noted above, and yet Hunt spent a whole Me i No. II on
"duality", followed by Blumenthal (Hunt's student) nd Getoor's book.
Besides thdse holes you saw in the semic~nductirs e you can list
some physical applications of your reversing in Part II. After all
physicists think facilely of those doings from high heaven down to earth.
and TALK







Prof. Walsh June 16, 2003

John: I sent you a fax yesterday and to-day. The latter
replaces the former.
I have forgotten if I have already told you:
(1) When I said Meyer preferred Claude to ..., I meant you.
I found a mistake in Claude's book, you gave a counterexample but
Meyer cited his --- something like this, not well remembered. After
all both Claude and you were then there and must have done similar
things, "no problem".
(2) It would be good to show a double reversal in your stuff.
I did that in some place but remember you may need a preliminary
condition such as the reversal time is finite.
(3) In Part 2, where reversing time is a big deal (not in
Part 1 where it is only casually mentioned), some solid history
should be given in Notes or otherwise. What Kol. did c. 1930 is
NOT reverse, it is dual. A GOOD history is given p.271 ff. my
M. C. SecOnd edition [if your poor library does not have it, get
one from somebody and READ IT.. I told you c. 1968/9 about my
reversing the minimal chain from its first infinity and said I
was unable to generalize it because I need a density denoted by
li(t) there. You had the bright idea of prolonging the life time
3-exponentially and the rest goes just as in my special case. C'est
la vie. The MMP is your great contribution, is it by that essen-
tial limit? I forget. Need to read my footnote on p. 44 of my
Random Time to appreciate this --- maybe I have said so in another
fax --- wonder (f Meyer ever saw it. I did not send him a copy
partly because he told me he was entirely out of math, and part-
ly I could not ask him to send me a can of pate (not goose, too
heavy for me now) as pay-off like your bacon. No kidding: your
new part under Time Symmetry deserves a brief history (a la
Bourbaki gxxz if you can. It should be "fun".


44/ Jlu,(

Zef
~~~~~~^t^?







Prof. Walsh
June 19, 2003

FINAL Note et I

John: I enjoy your expansive response but must warn you about the
Notes you write (!) in your part about TIME SYMMETRY The book is for
new readers who don't know a thing i4 --e b.3-^I Pe MN 1 r
_- 4jsudlaL tL L ,~4-r eicT TwoA%(2 b b M par phs would do. Who cares
for all those "credits"/ I just read again the first page till the top of
second page of ou an J=17r) article. DO READ IT, word by word if need be.
7 $*arted with IME mmetry and ewhi. the duality assumptions of the
Japs. I was too kind to Meyer not to cite his unsuccessful account in
the Seminaire, which is now cited on p. 44 of my pound-of-bacon book. You
should find Meyer's artucle and perhaps quote him directly about the un-
solved problem of time homogeneity of the reverse (I forgot what he said so
you must check it). Then we can simply say that problem is now solved here
in Chapter ?, Section ?. Basta. YOU (not I) can just refer to our paper
for more nitty-gritty history --- who cares now? After we prove the path-
by-path reverse ofy a Hunt process, of course the reverse i left con-
tinuous but that is not enough and now you can "brag" about essen ial limits
MMP (I already did it for you in N'ja eene ?). That is enough history fdr
-
the innocent reader. If you want to add more gossip that will be fine but
I won't overdo it: Nobody understands nor cares. E. g. what was done in
discrete time chain with a stationary distributionFis Neanderthal: we now
think it was not Human. I forgot but just saw that I even citFe Ed Nelson's
Chicago Ph. D. theses --- I was there. Even Ray's paper is in my Bibliogra-
ohy in the'first ed, *f my MC book, 196o, and in my I~dex.
I hope this is my TnaI effort to contribute to the new edition, except
a very brief joint Preface, in which we can just say that t consists of
two parts, etc. As usual that can be delayed until galleyask Ina. [My
old Preface must be reprinted with one misprint corrected]J
Postscript (June 29). If Ina wants to cut, here is an easy way.
Just erase all those folksy style in the MS. suitable maybe if you talk
in classroom but unnecessary and unhelful in a TEXTBOOK: GRUNDLEHRENT
This was noticed right at the beginning of your MS but I did not comment
on it since you are your own free agent --- before Ina ... Ha! A good
editor can do it for the inexperience author. Have you written a BOOK be-
fore (lecture notes do not count). I have written OVER a dozen.







Prof. Walsh .
une 20, 2003
John: cTt +4(;
(1) Have you READ the artle about retourement du temps by Andre
in Seminaire 1968. I found it and read it again. If you have not read
it yet, do so before writing your history. I wouletstart it with a quote
(in French) from our dead friend. .He was honest '( as others were not).
(2) At least I hope you have read the FIRST page of our article but g
do not stop unyil the top lines on the second page. I (Moi) would not
give'the Japs [until their government makes formal confession and "apology"
as the Germans'did to the Jews they will remain Japs to me? of course I do
not extend this to some mathematicians I know) --- repeat: IOULD NOT give
those Japs mentioned on the first page too much credit because they be-
gan with DUALITY hypotheses (check those; I forgot). If you begin with
duality what do you want to reverse for? Cf. Blumenthal and Getoor etc.
Remember good old Gil not only had to assume duality but even further
[Cartanlike, I forget] specialization for his No. Memoir --- which we
did not go through. Remember we went through a large portion of'his No.
I I bet Gil did not assume/duality in his '(last) paper about Martin
m~;W1 boundary, but that is for discrete time --- pas de problem lA. WHY did he
1/t not do it in CONTINUOUS TIME? I did it in my Martin boundary note in
continuous time (discrete space is no big deal) and that wa+MY MODEL for
reversal. I think I told you so in an earlier fax with that density of
the reversal time li(tyr-eam glad to see it in y&ur MS: the key to .
(3) It just occurred to me: in the continuous time I suppose we
can also have a 'stationary distribution": e. g. for Brownian Motion it
must be the BAgr-Lebesgue measure.: no problem with infinite measure as
Derman had already done for countable chains [that was at that time quite
.; something : even Feller put it in his book. The Russians never got it]
193 Now if there is a stationary distribution for a general "Ray-Hunt-Feller"
process will it then be trivial to reverse as Kolmogoroff did (you cited
it: but did you ever read his paper? I didO] I do not know the answer.
(4) W&t really beats MEps I know of few if any good applications
of reversal!l That woman in Cicinati once said she had some, but it was
extremely special/. Do some if you can, not too late. The test of the pie.
Of course I have'many great applications for "LAST EXIT TIME", even 'lectri
city but that is in Part 1. Ciao.







5eAt ?
1/20 June 16, 2003

John:
I sent you by fax my amended Bibiliography and Notabene, second
version. Please have this inserted and tear the first version sent one
day before.
Your "history" of reversing is all wet. Kol's is not reverse:
it is dual. I think it is given in Feller with extfon to the
denumerable case ddne by my Ph. D. student Derman, in Feller's
book. See s St 'm) n ,C.
(REEAD p. 44 of my RANDOM TIME, the footnote. Did you? I was
kind to AndrA there.ff you can find a copy of his WITHDRAWN artici
le (I have a copy he would rather forget). He did not knww if
a (Hunt) process reversed would still be homogeneous. See if
he was honest to say so? There are perhaps "hints" of this in
Hunt's (last Paer on Martin for DISCRETE time, but I do not
remember whether he was able to show that the reversed chain
(from a last exit time) s stationary transition amely
"homogeneous". The Japs Nagasawa and others did reverse from
ceratin "reversal" times specifically cited on p. 1 of our
1969 paper. But I do not remember if they proved in any
manner the stationary transition of the reversed "portion".
-f you are interested you should ,reat it and tell me. As far
as I know, the VER-,FIRST TIME that a reversed process has
stationary transition is given in MY book on Markov Chains,
where the reversal time JS the FIRST infinity of the "mini-
f remal chain" -- READ IT. It may be evlein the first edition
if that'siotly one you can fnd.i The time reversal in conti-
nuous time was regarded either as. an "obvious" thing or as
"who cares"? You are too young to know that. Your great
S. 8M teacher as-far as I know, did not take it up. He did re-
0"41 verse at a conatnt tl see MY Lectiures> but I do not recall
+40- any other diddling by him, nor by the French schooluntil
Yotr P- t OUR (wow) paper. Read Meyer's re-write -which I eecall has
s3mey merits (maybe you make use of these in Part 2 --- I did
not see ~uYf"l& not read the details anyway). Were you at
A
the Santa Crr-z confab where Azema. and 1i- fe, -.l.
and dead Bauer were all there? I is W"W'y- .
KQ\I





TRANSMISSION VERIFICATION REPORT

TIME : 06/15/2003 06:96
NAME:
FAX :
TEL

DATETIT 806/186 V:!
FAX N"AME 10106340 0826 9581523
DURATION 00: 00:
P (S) 01
T
ANDARD
CM





I think it was supposed to be about REVERSING TIME, but as we know
degenareted into all kinds *-atuff. I did jiot remember if it has any
record though it was partly supported by NSF, -throgh my and a special
grant. The German(s) complained about their shared dormitory rooms
and asked me to protest. I did by telephone on the spot, spoke to XXX
[Forget the known name]. You know how such things went. Hsu Pei (do you
know his book in AMS??] ~gib- me a ride,....
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Prof. John B. Walsh: La voiiB1a- votre service. Je miarrete.

Pc 1/6/2003

APPENDICE AU CHAPITRE XII / 1

RETOURNEMENT DU TEMPS




Nous avons vu au n* 3 que le "retournement du temps" preserve le
caractare markovien des processes. Malheureusement, il ne preserve le plus
souvent aucune autre propridt6 int6ressante : par example, supposons que 1'on
effectue le renversement du sens du temps sur un processes (Xt) admettant
une function de transition homogene dane le temps : on ignore si 1e processes
obtenu admet une function de transition ; mame si celle-ci exiate, elle ne sera
en g4ndral pas homog&ne dane le temps. Enfin, la function de transition du prO-
cessus retournd (si elle existe) ne depend pas seulement de sl function de tran-
sition (Pt) de (Xt) mais aussi de la loi d'entr6e de ce processes. I1 s'agit
done d'une operation peu satisfaisante.

Les re6ultats donn6s ci-dessous sont el6mentaires. Nous reviendrons
plus tard sur le problnme du retournement du.temps, dans 1'un des derniers
chapitres de ce livre.

23 Consid6rons un processes de Markov (Xt)tE T defini sur un space
(C, F,.), admettant une function de transition markovionne (P, t) (non n6ces-
sairement homogene dans Ie temps), et une loi d'entree (vt) Nous laisserons
au lecteur Ie soin d'4tendre ce qui suit au cas oi lea measures vt, au lieu d'etre
des lois de probability, sont des measures 0-finies sur l'espace d'6tats (E,p
(voir le n* 15)

Supposons qu'il existed sur (E, F) une measure positive finie 1 ,
tell que les measures P t admettent des densit6s par rapport & 11

P t(x,dy) = P, t(x y) 11(dy)

mesurables par rapport au couple (x,y) et satisfaisant & une "relation de
Chapman-Kolmogorov" tr&s precise

_Pr, ,(x, y) 11(dy) p, t(y. z) pr, t(x, Z)
Hey. I warrnL u iask yuu a question uonreilaed tu the book but I HOPE
with your renowned intuition (even Andre knew) you should think about
it a few minutes and give me an ANSWER. It is explicitly stated in my
Note about Martin boundary for Markov chains, 1962. I proved a result
there sharper than needed for the Doob-Hunt Martin boundary(for disaie-
crete time) which I thought led to a finer boundary invarient under the
shift? Please tell me if there is really something there YOU DID NOT
know??? 41 years have passed without anybody giving me an answer.
All this latest digging up the (my) past made me wont(er. R.S. V. P7.






- 18 -


APPENDICE AU CHAPITRE XII : /2

RETOURNEMENT DU TEMPS





Nous avons vu au n* 3 que le "retournement du temps" preserve le
caractere markovien des processus. Malheureusement, il ne preserve le plus
souvent aucune autre propri6t6 interessante : par example, supposons que l'on
effectue le renversement du sens du temps sur un processus (Xt) admettant
une function de transition homogene dans le temps : on ignore si le processus
obtenu admet une function de transition ; meme si celle-ci existe, elle ne sera
en general pas homogene dans le temps. Enfin, la function de transition du pro-
cessus retourn6 (si elle existe) ne depend pas seulement de la function de tran-
sition (Pt) de (Xt) mais aussi de la loi d'entr4e de ce processus. Il s'agit
done d'une operation peu satisfaisante.

Les r6sultats donn6s ci-dessous sont 641mentaires. Nous reviendrons
plus tard sur le probleme du retournement du.temps, dans l'un des derniers
chapitres de ce livre


23 Consid6rons un processus de Markov (Xt)tE T dfini sur un space
(0, F,P), admettant une function de transition markovienne (P ) (non neces-
-w s, t
sairement homogene dans le temps), et une loi d'entree (vt) Nous laisserons
au lecteur le soin d'6tendre ce qui suit au cas oi les measures vt, au lieu d'etre
des lois de probability, sont des measures a-finies sur l'espace d'6tats (E, E)
(voir le n* 15)

Supposons qu'il existe sur (E, E) une measure positive a finie f ,
telle que les measures e P admettent des densities par rapport a T
x s,t

Ps, t(x,dy) = Ps, t(x y) (dy) ,


mesurables par rapport au couple (x,y) et satisfaisant a une "relation de
Chapman-Kolmogorov" trbs precise


pr, s(x y) (dy) PS, t(y, z) = Pr, t(x' z)






University de Strasbourg
Seminaire de Probabilitds 1969/70


t LE RETOURNIEMTENT DU TIPS, D'APRES CHOU7G ET WALSH
Sorpar P.A. Meyer

W /O CTHUG et WALSH ont public recemment un article )qui fait fire
5 de tres grands progress & la th6orie du retournement des processes
On 7V) markoviens. Nous exposons ici leurs resultats, avec des methodes un
peu diff6rentes, et en nous bornant au cas des processus fortement
markoviens.

1. FORPE GROSSI;T.S DU TI) OREI3 DE RETOUIETBM ET
Nous nous donnons sur un space d'etats E, homdomorpho a un couo-
espace borelicn d'un space nmtriq.ue compact, un semi--groupc sous--mr-
kovien (Pt) qui satisfait aux 'hypotheses droites" : 11 admet une
r6alisation continue a droite, fortement marlovienne, tell que les
functions p-excessives (p>O) soiont prosque-bordliennes et p.s..
continues A droite sur les trajectoires, Nous const.ruisons la r'ali-
sation continue a droito canonique de ce semi-groupe, t valeuire dans
EU{f : 0, fo, X. ........ C ous designerons par (U)pO> la rEso.-
vante, par U le noyau potential.
Nous choisirons une loi initiaile X, ui restera f:.- d to
l'expos ( les probabilit6s de transition des processes retcurn`.s
d4pendront de )., et nous desinearons simplement par P ia loi pA^
par p. la measure potential. U. .i-n que cette rmesure n- soit pas tou-
jours a-finie, aucune precaution sp;cciale nest ndcessc.ire pour lui
appli.quer le thdoreme de Pubini, car est some d'tune suite de me-
sures bornmes.
Le processes retournd (Xt)t>0 est d6fini de la man.isre su.vante :
si o si t>(e(w), ou ()-co X^ C
Ces formules, ddfinissent, contrairemoe.. & Iabitude, un processul
,* 1 ,. contil, ^ .. *ae.c L'une des originLali 6s du travail e CiUNG-US
. consisted .prie,-; nt a utiliser le rtourne tel qu'il est, sans
S (o r,-e L rkov oce~si A a Kath. 123, 1970, 225-251.

r/y ev^d:,, A e C A. /A ,.,O/)er ..;.../- .-....a
.2 /970 -T Ave r Orse e5/ of-,e r e. .
1a' P l e4(lf 7tva -P^r A -l






Prof. John B. Walsh: La voici18a- votre service. Je mfarrete.
21/6/2003

APPENDICE AU CHAPITRE XII

RETOURNEMENT DU TEMPS




Nous avons vu au n" 3 que le "retournement du temps" preserve le
caractere markovien des processus. Malheureusement, il ne preserve le plus
souvent aucune autre propri6et interessante : par example, supposons que l'on
effectue le renversement du sens du temps sur un processus (Xt) admettant
une function de transition homogene dans le temps : on ignore si le processus
obtenu admet une function de transition ; meme si celle-ci existe, elle ne sera
en general pas homogene dans le temps. Enfin, la function de transition du pro-
cessus retourne (si elle existe) ne depend pas seulement de la function de tran-
sition (Pt) de (Xt) mais aussi de la loi d'entr4e de ce processus. Il s'agit
done d'une operation peu satisfaisante.

Les r6sultats donn6s ci-dessous sont 616mentaires. Nous reviendrons
plus tard sur le problem du retournement du.temps, dans l'un des derniers
chapitres de ce livre

23 Considerons un processus de Markov (Xt)tE T dfini sur un space
(0, F, P), admettant une function de transition markovienne (P ) (non n6ces-
SP t
sairement homogene dans le temps), et une loi d'entr4e (vt) Nous laisserons
au lecteur le soin d'6tendre ce qui suit au cas oi les measures vt, au lieu d'etre
? des lois de probability, sont des measures 0-finies sur 1'espace d'1tats (E,_E)
(voir le n" 15)

Supposons qu'il existe sur (E, F) une measure positive a finie 1 ,
telle que les measures ePg, t admettent des densit4s par rapport a TI

Ps,t(x,dy) = p, t(x, y) 1(dy) ,

mesurables par rapport au couple (x,y) et satisfaisant A une "relation de
Chapman-Kolmogorov" tres precise


SPr, s(x, y) I (dy)Ps t(y, z) = r, t(x, z)
Hey, I wanv Lu ask you a queuLilon unrelaLed to the book but I HOPE
with your renowned intuition (even Andre knew) you should think about
it a few minutes and give me an ANSWER. It is explicitly stated in my
Note about Martin boundary for Markov chains, 1962. I proved a result
S there share er than needed for the Doob-Hunt Martin boundary(for di~oa-
crete time), which I thought led to a finer boundary invarient under the
shift? Please tell me if there is really something there YOU DID NOT
know??? 41. years have passed without anybody giving me an answer.
All this latest digging up the (my) past made me wonder. R.S. V. P-








June 28, Saturday, '03


(1) I am surprised you have not found the Sem in my footnote
on p. 44, told you at least three times. Anyway only the first page I
faxed you is needed for your "historical notes. 'Yes, make it short
AND so begin by quoting your dead friend there. If a dual semigroup
is assumed, jap or else, that's not kosher. Q;, Cera;f r dTJi I
(2) Suppose the chain has the stationary distribution (w ),
do the Exercise: t edy d"cVyf e siec
P{X nl=i X =j} = ij DEF p*j ---
Sn- 1 n --%-4 V^a Lene-
Prove (p i.) is a proper transition semigroup. Don't sneer: add this
as Exercise No. 1 in your section on Reversing TIME. CREDIT it to A. N.
(?) Kolmogoroff (not -v in 1930). No joke here: the book is for stu-
dents (you tend to forget this) and they should do this Exercise.
(2) In more than one published letterStojHerald Tribune
(now defunct and bought by NYT, of which I am a shareowner),l'told
a little of the story they did when I was a student. Glad to hear that
Joke had no such memory. \Al.
I have thought of drafting a few lines for our joint Preface:
This book consists of two parts. The first part, Chapters 1 t
to 5, is essentially a reprint of Lectures .... 1982. A number of
minor misrpints have been corrected, beginning with the misspelling
Ve T[rinot dte to jaJ in the preface. A few additional remarks are insert-
Zed < &(references and
ed> at random, without any iRxmxfa x~i attenpt at
omerA
retrouver Ie temps. That formidable task is left to the new Jecond
author who wrote the second part of the book.

YOUR TURN, je t'en prie* Here is a final suggestion: if Ina
insists on cutting, some five to ten percent can be saved simply
by omitting your expansive chatting stle (as you do in the class-
room2 but maybe too folksy in VTEXTB 0K: Grundlehren. Certain-
ly no Frenchy or German PROF. talks like that? Not Andre,
n t eU ag YgU tRdwR Ye 'frst gr g51o hK nita-Wata-
sand s dollars (worth around 400 jap yen then ) to hal- a
dozen of them beginning with Kiyoshi: moi do not discriminate.
Some (e. g. Kunita) did not even submit a "Report" required by
AFOSR ( as you did?). Voici le temps retrouve.


Prof. Walsh






STANFORD UNIVERSITY
[.AOr S\ < "TANFORD, CALIFORNIA 94305-2125
DEPARTMENT OF MATHEMATICS
(650) 725-6284
FAX (650) 725-4066
June 30, 2003
Prof. John B. Walsh

To-day is the LAST day in June: read the contract with Ina.
Two paragraphs should do for YOUR Notes on TIME SYMMETRY (your
title).
(1) I jkst knew all along that you never read or saw that page by
Andrd faxed to you as "guide" (NOT Pis QuAde 3 years later). That is
why I took a lot of trouble to dig it up and faxed it. I am amazed you
still asked waeit i't t I asked you to rad..
p. 44. This must be the fourth crow. Je quitte.. ---
(2) Did you not tell me that Meyer's expose of K-W duality made
c \.196 .
you go to sleep (in Strasbourg classroom?). There is a "no sweat" way:
simply Eei2r the "interested reader" [I wonder if any] to pp. 225-6,
Acta Math. v. 123 (1969)1. Ca y'est. ;i m 5 Bl togrp-Ay ho.
(3) If Ina objects, an easy way to reduce the page number is
to leave out all those folksy chats which may be good classroom style
but not necessary in a TEXTBOOK: GRUNDLEHREN. Am sure Ina thinks 'o but
too polik to say it. Want a few examples?
(4) I will send a draft of half of the new joint Preface, which
will begin like this: ce oY!
This book consists of two parts. The first part, Chap-
ters 1 to 5, is essentially a reprint of "Lectures ..."
1982. A number of misprints and mistakes have been
corrected, and a few additional references and remarks
inserted. The latter must be regarded as randomly
selected since ..... l CL o U e coh;nueo) -20oyeorss q
The second part ... [c'est a 480 I o InM)*"
ia Nota Bene solely for you: thousands of US dollars
(worth about 400 Jap-yen -at the time) were paid by AFOSR from my con-
tract to most of the persons mentioned on p. 225 (loc. cit.) ---
.: CrMi._'-" there. Some never turned ia Report required by AFOSR
Did you have any "visitor" on your canadian grant?

k-, Zc.






Prof. Wqlsh
June 30, 2003

SS&m;,a e /~ ig I guessed all along that you never read (or even SAW) that Meyer
article which I sent by fax. Now READ IT. Yes, one or two matter-of-
fact historical Note &n your chapter/section on TIME SYMMETRY suffices.
aTs Quote Meyer (sent to you) verbally and say where the homogeity of the
A
pathwise reverse IS proved in your part. To avoid all "credits", just
refer to page 1 to (top of) page 2 of our article 1969, written by
one of the authors Oy the present book. No need to go over all thosi
names mentioned there, unless you want to put in your teacher and my
classmate (Gil Agnew Hunt, did you know), for LE TEMPS RETROUVf.
I willtsend you in due time a draft of our joint Preface, very
short, leaving the second half to you to fill in.
You can easily cut out some five to eight percent of the Dvaes
in Part 2, simply by dropping the folksy style verbiege (I found
10
spomeweeks ago in the first batch of MS you sent me). Such chats
may be fun in your Vancouver or Skrasbourg classrooms, (provided
they understand your Irish slang?) but not necessary (nor suffi-
cient) ina printed TEXTBOOK: GRUNDLEHREN. I estimate this may
satisfy Ina's demand for page restriction. SA io peld' -o iell yoci
Our Contract specifies delivery in June (perhaps first not
last day) To-morrow will be June 30. I leave this to Ina and you.
I will supply in due time a short draft of half of the new Pre-
face, as follows (subject to revision)t
This book consists of two parts. The first part. Chapters
1 through 5, a6e essentially a reprint of "Lectures from MP to
J M44 Y4af4
BM": 1982. A number of misprints have been corrected and a few
additional refences and comments have been inserted. The late
ter must be regarded as randomly selected (by the first-named au
thor of this new book), since much has transpired during the
over twenty years since his original w g based on classroom
Cha, (o delivery as lectures. The second part is by the second-mamed
ro ? author ........... PHa. your turne~e feel free!
bo 1 Apologize"'about the obvious lack of coherence--=wema relevance
and reference between the two independently written parts.]
P. S. Didn't yo" tell me Meyer's write-
up of Kunita-W. sent- *. '









JULY 1, 2003
Prof. Walsh


John: Hey our cohitract says June. When are you delivering?
Sure two paragraphs of Notes on your first section on reversing
are enough. Refer the "creditors" to pp. 225-6 in Acta Math., 123
(1969)in my Bibiliography. Saves space. Precede this with Meyer's
page I sent you by fax: he knew what he was talking about, hein? I'm
flabbergasted that you asked me%' sat document --- on. &44 ; this
must be the 4th time I told it.
If Ina grumbles here is an easy way to cut pages: just drop the
folksy verbiege which may be cozy in a classroom but this is / GRUNDLEYR-
EN. A good editor can do it for you. be
Let me know when the Preface is duef/ I have already composI O
the first fea sentences:
This book consists of two parts. The first part, Chapters
1 through 5, is essentially a reprint of "Lectures ....",
1982. A number of misprints and mistakes have been cor-
rected; and a few references and remarks inserted. The
latte must be regarded as randomly selected since twenty
some years is a long TIME to retrace steps ....
The second part is written by a new author ....
essentiallyly new /),
Your turn' Anticipate the question: "What's the "relationship"?
FIN


Anybody reading your MS?









Prof. Walsh, Math.


Hey what's up? I am pretty sick and can't do a thing, but
let me know what Ina and you are doing. Don't bother with anything
mathematical, just the publication of the new book.

No cheers at this moment.
0

Onfy Fax, no other.























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July 28, 2003







Math Dept, UBC: Please phone Prof. John B. Walsh to come and get
this.

John:
Ruth Willaims was here and apparently she saw your big bunch.
As I said before, it's between Ina and you, but you are late.
I will do nothing except add joint preface at the page proof-ing
stage (or earlier if abs. nec.) and scan my own revision, only.
When I write a textbooko, such as Lecture~is, inly put in stuff
I was thoroughly familiar with and had taug7h in class (not in
all details but approximately) many times i e-las. This was -
the case woth my Lectures, ergo the name. I river tried to do
"new research" or tread on uncertain grounds. This was also
G. H. Hardy's way who wrote couple of marvellous books I have.
Writing a book is not doing research.
Most impSrtant news: there is a good chance that I will have a
new and enlarged edition of my (unique) Princeton Press 91
page book written largely by Paul Andr4 who attend my semi- k
rY Cnaire (before you got to your mill there). That would be a
S suitable place for your general stuff that can illustrate my
boundary for chains, aid I should welcome a brief supplement by
you since I have seen some of your examples in this connection.
That will be up to you: no co-authorship and little money.
I mention this possibility which may help you to decieeWto dc
"*T your Vartin stuff etc. But the examples should be inclu4-
ed in your part II.
C'est la vie.


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04 5- 22 6 o y 2
Prof J. B. Walsh, URGENT
9/8/2003
John: l1?S5 f / e l CCc
Now I know from Ina that she hasn't got your MS with MINn .r She may
needya title soon to be announced, here it is after long deliberation:
Markov ProcesS, BrIonian Motion, Time Symmetry.
This is in the order of the new text. Any reader who wants only to kn go BM
will be totally lost in your portion so don'ilet's ki44hm. "and" is not
needed and icte Bqwill insist a comma after BM, as I do in my Green...
Next I have written the first piece in a NEW PREFACE, any suggestions??
9 ^ ^ ^ ^U ^ e4&"1, w^t C t)
This new gd-ti i~ under a new title is a considerably proloned
v4A- of [Lectures from MP to BMW, 198?. e grig text hres-e -
(Corr ectlones an additions have been made in
The second part from p.?? on is completely new and hithertofore not readi-
ly available for general readership. Historical Notes explain certainn
motivations and connexions with the first part., w'~-ah can be read without
knowledge of the second. [Lkhopetbat( There is only Doob's
S 7" fixed t reversing for BM ---did YOU
Cy V 5E .see??]

The a.et is entirely up to youL If, and dfly if, you need or want to cdn-
sult me! you must send it very soon. Je suiis tr ,s fatigue
You se he whole MS to Ina, but I will send with your su the
.4E9iga PRE1 T E, as soon get tkew q quick. fi
Fax kf any questions? Mail is too slow and I do not telephone these days.
Yuke ma be interested to know MY new Elem. Prob. just out, earned me
some 3,CO aleady, taxable. not to '5a "arid's cut. I do not need it, will
just get tw me aL winei the 2001 Fleurie (ever heard?) is not worth it,
Never kno when you-guys get an URGENT fax, But retlun one as soon as y>y
get this: your good secretary CANDO Pe m.

"L





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"(" /Prof. John B. Walsh 10/7/2003

On p. 165
who's is wrong spelling
In the last-but-one paragraph, what is stable continuous time larkov
chain"? Neter sen this name anywhere before. Please explain the definit-
tion and spell it out for the reader. Sendga fax for the correction
4WRgy#jb# to be made, before sending it to rpitt.
p. 167: ????? to fill in I be om Ma i
line -4: p fpr P AI h' + I pr4f.*+heoy.
p. 166, line 5, after my name add reference to part l, LOOK IT UP
(since I had shown it to you), in t heNtas i MY NOTES on p.74. I shall
add, if I have not done so in the amended copy sent youi.*, my story
that I realized its moderate birth only at the last stage of proof-read-
ing the galley of that paper. Please check to see if I have done
so in the copy you have and report back. I saw you even told your
story of marginal notes on some MS, whigh seems a bit too much. Who
knows what? see p.164, sec.2.5 .-J t keey A c0y.
Typing on both sides of a page for a book should be forbidden by law
No page No: p.165. NOTES:
Sec. 1.5: may be "innocent" to you, not to Doob (1942, 1945), Feller
(before Doob cited above), Levy obviously did not find it so. If the) at
runs continuously in R and yet at each preassigned tine t it can only-h
take (say) an integer value, WHICH IS COUNTABLE (and discre%), Cantor
who invented contnuum/ denumerability (remember the Greek rationals:L.a
and their problem of the square root 2T READ EUCILD, not too late)?."
any innocent student with a certain intelligence would wonder "How
does it GO??". Even when al states O4 the chain aretag a f t
M'twhat you wrote on p. 165 cited above on this sheet) are all stable
[Did I not cite another example by Levy in an earlier fax---did Yb)READ
his "lips"??] .... By the way, let me hope when you wrote there on p. 1
p.165 loc. cit you do not mean "all states are atablel for your new name
"stable ... chain". "continuous -time" should precede your UNDEFINED
"stable chain". Is it not precisely defined in your reference to my [Fl.
Sorry I refuse to check it --- you do it. The rest of your MS in
faint handwritng is illegible to me. Do not bother to send it again.
Just let me know when I have to write my part of the Preface, and ye
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5/10/2004 10:41 PM FROM: Fax TO: 101594516508570208 PAGE: 001 OF 001


Hi, Kai Lai,

Meyer died on January 30, 2003.

As far as I know, Hunt's middle initial was only wrong that once. But don't worry,
there are lots of other typos. I find some every time I re-read it.

I'm tempted to tell everyone that when you read my part, you only found one error.

I hope all's going well.

Cheers,

John




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