Paul-Andre Meyer, 1963-2000


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Paul-Andre Meyer, 1963-2000
Physical Description:
Meyer, Paul-Andre
Chung, Kai Lai
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Box: 1
Folder: Paul-Andre Meyer, 1963-2000


Subjects / Keywords:
Mathematics -- History -- 20th century

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University of Florida
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Full Text

Dear Chung :
Thanks for your letter. The fact you have read that chinese book
pleased me very much, and if we meet at Durham, I'll ask you some
questions about it. Don't worry, by that time I'll be very,happy if
I have read page 10 1

I entirely disagree with your comments on Dynkin vs Maisonneuve. They
might well bring you a suit at the Committee for Un-Chungian Activities.
All the time you were at Strasbourg, you complained that the Continuous-
timeMarkovers proved theorems that were like those the Markovehai-
ners needed, but with a c hypotheses just to have the proofs work ,
and which were either false or impossible to check before the work was
finished. Now look at Dynkin :
standard process in a nice space
excursions from a closed set
absolutely continuous kernels with lower s.c. densities

While Maisonneuve
right markov process ( nothing from the left )
excursions from any homogeneous set
no absolute continuity

This is exactly what you need : you turn your bad chain into a right
process using a ray compactification and still you can talk about
excursions from the boundary though the boundary isn't in the same
pv&m*Ss, but in the left ( martin ) compactification. I agree that in
the case of chains you can get absolute continuity !
Remember Same results from different hypotheses aren't the same
theorem" Does there exist a quotation from the Classics with this
meaning ? There should be.
Another comment : your criticism of Maisonneuve really concerns
Getoor-Sharpe : Maisonneuve's contribution is the markov property of
the incursion process, and the unification of the theory around that.
I agree that Dynkin's paper is beautiful, but he's an analyst : Pitten-
ger-Shih, then Getoor-Sharpe, were the first ones to work under stochas-
tic hypotheses and our work on this subject consisted in reducing
these hypotheses to minimality, without changing anything from the
ideas of G-S. Karoui-Reinhard tried the same thing on Dynkin ( succes-
sfully ). Now Maisonneuve has a very short proof of G-S.
About Kailath-Segall : I'll be content if the revised version has
the rectifications I sent qo The paper will then be readable.
VyTK4 cb 1znfton

A, RNL\.

Dear Chung :
I think this is a proof. Let me say that a function is nice if it is Borel bdd
and pU f->f as p->co lower nice ( upper nice ) if f of an increasing sequence of nice is lower nice. U is a Feller resolvent, so
continuous => nice.
Lemma. Let f be p-excessive continuous. Then for g Borel bounded >0 fAU g is
Proof. True for g continuous. Assume g is use, so g=inf gn gn continuous
(gn ). Since fAUpgn and fVU gn are continuous, they are nice, so fAU g and
fVU g are upper nice. Since their sum f+U g is nice, each of them is nice.
Assume now g is Borel bdd and fix x choose gn use < g increasing such that
U gn(x) tends to U g(x), then fori every n lim qU ( Ag)X > lim qU (fAU g )x
= fAUgn(x) Letting n->oo we get the inequality in the good direction.

Lemma. Let f be Borel bounded positive, then U fAU pg- is p-excessive.
Proof. Start with f continuous and proceed as above.
For the theorem now use approximation by sup of potentials.

What << people >> say about Yan or Zheng is nonsense. If said in your presence
please react violently. Yan is a quite modest person, very hard working and
strong willed, if he does things in the Strasbourg way it is by his own choice,
and I never wrote a paper for him. What he does isn't great mathematics, but
every single thing is a produce of his own mind and brings as/improvement- in
understanding some subject. Zheng is entirely different. He is able, I think,
to do great mathematics, but he came to me as a student ( which wasn't the case
for Yan ) and I had to teach him order and clarity in writing. So I rewrote
his thesis in complete detail, to show him that when one edits carefully a the-
orem ( even after one thinks it is fully proved ) one usually gets rewarded
in the mathematics themselves ( not just in outward shape ). All the ideas were
his, and he astonished me. Since I had rewritten his work, he forced me to sign
it ( which I never do for students ), so I forced him to sign the paper on weak
convergence ( the general one, not that on stochastic mechanics ) which we had
discussed together, but was really 90/o my work)
Let me also mention that I deeply estimate Yan. I consider that to hold respon-
sibility in administration of Chinese he-would be-a reliable- person. He has
a more solid personality than Zheng
VJ~JAJ IxiJ Vt kt A H N

Dear ivir. hung :

Your proof is quite correct ; mine was correct too, but less
elementary The point to check was the following : for any t for
any sequence tnt t, for any martingale on Oxd' : (Ys) = (E[YIFsxF']
we have a.s. lim Ytn = Yt a.s. iNow this is true when Y is equal to

XX' ( XeL1(n), XIeL1(0')). Thus take a sequence of such product
random variables YP which tends to x Y in L let (YP) be the
associated martingales, and let T be the set consisting of t and the
tn's. we have, according to Doob ( JYs-YPj being a submartingale >U)

AP{ sup |Ys -Yp < KE[ |Y-YPI
K is BS 2 or 3, I don't remember That is, sup IY -Ysp tends to 0
in probability. Extracting a subsequence, we may assume that it
tends a.s. to 0 i.e. we a.s. have uniform convergence of the
sample paths of (YP) to those of (Y ) on T, and the right continuity
at t passes to the limit. The argument is more complicated than
yours, may be, but X I have found it very
useful several times.
I am sorry, I couldn't check your proof of the theorem on
progressively measurable processes. After writing it for my book,
I put your letter and some notes together, and they somehow got
buried in my disorder. Bu when your second letter came I couldn't
make the comparison with the first one.
Very truly yours :

January 19

Dear Chung :
I am just as confused about MOT as you are, and I haven't kept
any copy of my preceding letters. This is why I waited so long
before answering you And may be what I'll write today won't be
Well, let me start with some shorthand terminology :
if G is any set of functions, bG is the set of all bounded-functions-
G is monotone means : the limit of any increasing or decreasing
uniformly bounded sequence of functions in G belongs to G.
G is uniform means : G is closed for uniform convergence
G is multiplicative : G is closed for products
G is infamous : G is closed for A
G is vector : is a vector space.
I always assume below that G l.
My little remarks now are the following :
1) If G is algebra and uniform then G is infamous ( the Weierstrass-
Stone argument )
2) If G is vector, infamous, and monotone ( or uniform ) then G is
This is less trivial, and isn't even stated in my book : note
that x can be written as sup an x+b for suitable coefficients
n n
an and bn ( convexity To get a countable family one may use Dini's
lemma ). This settles the monotone case The uniform case follows
from Dini's lemma.

3) If G is vector and monotone and infamous ( in particular if vector,
monotone, uniform and algebra ) then it contains bG (C%=bG 1 )

This is easy : jf1 1 has for its indicator lim g g=f Al

So these are the three essentially trivial remarks. Now what can
be deduced from it ? CP It
THEOREM I. If 0 is multiplicative, H/ector, contains/1,/monotone,
unif-~m, then H contains bC. That's the result in my book.
PROOF. There is some algebra in H containing C and 1, choose a
maximal one G and check that it is uniform ( hence infamous )
and monotone. From 3) it contains bG hence bC.
THEOREM II. If C is vector, infamous, contains 1,1 imonotone ( not
vector !), H contains C then H contains bC That's the result I
quoted to you.
PROOF. There is a maximal vector infamous G containing C Check it
is monotone ( hence algebra ). Then it contains bG hence bC ,

Variant : if feC => f A e_+, then you may just assume 1eH instead
of le0 Proof is the same you must just check at start that the
vectorinfamous generated by C and 1 is contained in H .

Well, I hope all this is correct, and usable. I hope that some
day a thesis will be written on." deformation of MOT" shawing that
all possible MCOT's form a p-adic Banach manifold of some kind, whose
7th cohomology group with coefficients in the sheaf of 2 isoO .
How uId'Karlin put his foot on Markov processes" ? ( I purposely
.omitted and potential theory" : neither Walsh nor I are interested
in that ). By the way, the same thing seems to be happening.., in
Strasbourg. Fernique now wants only STATISTICIANS.

Dellacherie told me he had been very pleased by everything in Stan-
ford. Your knowledge of Strasburgian mathematics amazed him.


March 13

Dear Chung :
I have something to tell you about mathematics, about which I'd
like very much to have your opinion. Unfortunately the xerox machine
is much more difficult to use now ( apparently it was so intelligent
that it kept doing copies quite alone in the dark, so they locked it )
and I'll have to tell you the story. That will be long...
First of all, I'd like to know your opinion about the recent
paper of Getoor-Sharpe. I am rather enthusiastic about it, because I
am able to understand it quite clearly, but John Walsh wrote me that
much of it looks pretty obvious to a Markovchainer. I'll first give
my notations for it
Xt,Pt,,=Ft,e ... are our usual friends : a right continuous

processes de Markov droit ( no left hypotheses at all, no absolute
continuity ).
M is a homogeneous random set, progressively measurable, in
]0,cO[ Otherwise stated, its indicator function (Tt)t>o satisfies
identically I s+t=I so In fact, I am interested only in the closure
of M ( in ]0,co [, so let me assume that M is closed. .1I set
D = inf { t> 0 : teM j, a perfect, exact terminal time
F =Ix : Px{D=OI=1
QtVp semi-group and resolvent associated to D
D = t+DoQt first entrance in M after t

Getoor-Sharpe consider only the case of M = { t : XteF} ( of course,
my F is their Fr ), but I am interested by the general case, since
Maisonneuve worked in full generality. I'll begin with the statement of
Maisonneuve's main theorem. Call M-incursion any w such that D(w)=+oo.
Then to each w and t you can associate an incursion it(w), which is
Q@t killed at D(Qtw)=Dt(w)-t Now if you want to describe the way
you go from M and return to M you need to know the-incursion at t,
also the length of the return time Dt(w)-t ( which usually isn't
determined by it( ) ) and the return point (XD (w)). Now
MAISONNEUVE's THM. The pair (Dt-t,_XD) and the triple (Dt-tXD ,iDt)
are homogeneous, strong Markov processes with branching points ,
both right continuous in the appropriate topologies, relative to the
family (FDt) ( not F of course, since they anticipate" on t ).
not =t
I'll denote the triple process by (7it). Now back to G-S. I denote
by M the exit set of M, that is the set of left endpoints of conti-
guous intervals to M. This is a ( not closed ) homogeneous, progressi-
vely measurable random set. I split it ( following G-S ) in two parts

-4= e
M~a = M fix{eF

Mb = M nlX.eFc

The second one is the trivial part : a countable union of graphs of
stopping times The first one is a progressive set without stopping
times,- a most wonderful object The main purpose of G-S is to compute
sums of the following'kind (\oa&<>iala% atc o ~e uu cJU -c CLab)
E[ Z ePgfDg e-shoX ds .]
g g
where (Zt) is a well-measurable process, and to deduce from it last
exit decompositions and conditioning results. The sum on Mb can be
deduced from the ordinary strong Markov property so the interesting
thing is
(*)' E[ Z e ge/... ]

and their result is as follows. One can find an additive functional
(At), continuous, carried by F .; for each xeF an entrance law Qt(x,dy)
generally unbounded, carried by Fc such that, if one sets V (x,dy)=
fO e-PtQ(x,dy), (*) is equal to

(**) EP[ *OO p(xth)ztdAt ]
Well, the first thing I'd like to know is your opinion of this result
of G-S : is it a trivial result from the point of view of Markov
chains ? Of course, G-S contains much more than that, but this is the
main lemma. By much more, I mean the conditioning part, which contains
the most subtle Laplace inversion I've ever seen.
Now, my own work begins here : since I have read Ito's Poisson proces-
ses, I forget about balayage, and notice that (*) is a particular case
(+) EJ1[ Z coQ ]
geMN g g
where c is any measurable (!O) function on 0 :. (*) is just (+) with
c = D h(X )e-psds From a little work using G-S, I can deduce the
0 s expectation operator of the
following. Denote by E the. Markov process; with (Qt) as transition
semi-group and the entrance law (Qt(x,.)). Then if c is relative to
the killed process at D that is c=cokD ( killing operator ), (+) is
equal to
(++) E'[ / Z E Ec]dA ]
This is pretty intuitive, and I'd like to know whether this 0*&t&
for Markov chains too. Now what happens if c isn't killed at D ?
Intuition suggests that the piece after D must first be conditioned
with respect to the piece killed at D, then the whole is computed by
(++), and the natural idea is to look for a Markov process with un-
bounded entrance law and (P ) as transition semi-group expectation.
operator Ex whose killed process at D is E (+) being equal to

(+++) EJ[ / Zt x[c]dAt ]

in the general case of non killed c But this I couldn't deduce from
G-S and here too I wonder whether this is standard in chain theory.
Then I found something : Maisonneuve had noticed that points in M
were jumps of the incursion process. On the other hand, the theory of
the Levy system is now developed in full generality for all right
Markov processes, and he had related the L6vy system of the incursion
process and the G-S kernels. I tried to do it the other way round,
and it succeeded 1 In that case, one has ready made a-finite measures
and must check they are Markov processes, but it isn't difficult on
to do, and here comes the wonder ( for me ) : the measures Px with
the expectation operator Ex do exist on n) and are G-finite they
are indeed Markov processes, with (Pt) as tr.s.g. and right limit x
at 0 but they are impossible to build from the entrance law since
they may have no entrance law at all or rather, such an infinite
entrance law that you can't recover the process from it. But still
more surprising for me is the fact that one can prove that process
starts from x, without using something like a u-process...
Well, from my own experience, every time I have received a long
mathematical letter from anyone, I have decided that wasn't the pro-
blem I was interested in, and suffered for 3 months before sending
a polite and evasive answer Please don't get bored with this letter,
if you don't want to think of the complicated things, just tell me
your opinion about Getoor-Sharpe.
Martin has become awfully fond of China, so I am trying to recall
my very old 2 years of Chinese, in order to teach him a little. I
bought him a brush and ink but he didn't want to use the ink because
it came in such a nice little box and he wanted it to remain new
for ever, so I'll have to buy him a second ink box that he may use.
With best regards,

vK, n

( 0 S

^^ee. ^s

March 23

Dear Chung :
.1I was glad to get your letter, though it contained much.criticism.
1) Of course I agree with you that this material isn't "open to the
public", and I hope very much I'll be able to open it, if life allows
me so ( I'm afraid of the amount of work I "should' do ).
2) I don't feel I have'ever generalized for the sake of generalizing, an
you criticism of Catherine isn't correct. One of the things mathemati-
cians need is computing and to know a computation may be done is a
-. big progress, even if you never read the complicated justification of it,
o My notes on K.W. may be less readable than the original ( I doubt it,
aC but I am may be the only one that read both of them ) BUT they contain
the version of the change of variables formula that works in all cases.
r- Catherine may be too abstract, BUT she gave the right formula for the
S exponential of a martingale. It turns out that engineers needed a theory
a of the stoch int. that would allow non locally bounded integrands ( the
Fact that we had considered only those previously just shows we weren't
g just fishing for generality ), and I givle it in my notes.
o 3) A principle with mathematics is that things you need are easy to read
H and look interesting, while things you try to read just for the sake of
.i learning are boring. I suppose that* the problem with those nasty com-
Sments of good people : they didn't need the material. And probably you
don't need it yourself ? So I'm not surprised you don't like it.
I like your pater on energy. it is the first paper that uses proba-
bilistic methods in the theory of energy, and I feel things are ripening
now for the use of them. I only wish* ( as I told you in my hurried
letter of last week ) you had written a longer one, because your use of
probability just to prove the technical lemma while you still use the
classical theory of energy isn't quite convincing ( it's like the first
paper of Doob on Brownian motion, in which he still used the whole
classical theory of potential ). Anyhow, the paper certainly starts a
new trend in probabilistic potential theory. I still haAq no time to
go into details. I was very upset by the clash with Springer, and I
couldn't think about anything else for some days : these administrators
are so stupid 1 Of course, in spite of my denial, it is true that GetoorE
paper was/expository, but it is good exposition for probabilists of the
version of the theorem on disintegration of measures they need, and I
feel anyhow we must fight for our freedom against publishers ( and speci$
lly publishers that pay no royalties ). So if you think that I should
rewrite these notes on S.I., please write Peters or anybody you may know
at Springer that they should rather reject my ununderstandeable stuff on

Ensembles Aleatoires, for instance, than these three papers which were
written to be understood. Don't be afraid you'll do any harm : after I
get this volume through, I plan anyhow to stop these seminar volumes.
I'm sick of the trouble they give me, and I'll just circulate the papers
I am leaving Strasbourg for Paris today, and mail won't follow me for
two weeks.-
With thanks again, and many regards,




7, rue Ren6 Descartes
T61. (33) 88 41 63 00
T61ex ULP (33) 870260 F
T616fax (33) 88 60 75 50


44dwQ? *

(~L'L~h2 &l-cI

n LkS ^ iJ



A) K



U' I CUc

" V ^ ^-%



UFR de math6matiques et informatique

Envoi de
Strasbourg, le

May 4

Dear Chung :
I think you shouldn't worry about the CR : the worst they can do is
to change the official date of presentation to the date they get the defi-
nitive text.
Your questions about measure theory :
(1) : I don't know, but I'm ready to bet the answer is no.
(2) : Let X and p. be sigma-finite. Then you can find f>0,( the same for both)
AUch'ohat ?1'=,'=f41fu.are bounded, On the Borel set If>es, both X and .
are bounded, and since every boundy-d measure on a borel set of the line is
Radon, the fact that they agree on compact sets implies they are the same
on if>el for every e, so they are equal. 7
(3) : putting masses on the rationals, it is easy to build a sigma-finite
measure on the line which gives infinite mass to any open set, so no hope
with continuous functions with compact support 1
There are many errors in the sans larmes, please be indulgent.
I don't know whether I'll go to China now : I sent my invitation more the
than one month ago to the french ministry and got no answer at all ( they are
supposed to pay my trip : may be they have no money ). It's a long time
since I have practically stopped doing chinese too : learning some differen-
tial geometry takes all my time...
With all my best regards,


-May 7

Daar Chung :
I am sorry you apparently took my letter too seriously : I never
meant yours had anything-really "nasty". However, your suggesting that
the rejection of the two papers, instead of being due ( as I believe it
is ) to the rigidity of some Prussian algebraist, was due to the cons-
cious ill will of some close colleague, was troublesome to me. And even
your mentioning the comment that I rewrite other's people papers, a
rather obvious statement of fact to which I can readily suscribe ( even
with the implication that the imagination hence is borrowed from other
people ), becomes quite hard in the context of ill will. I should'nt
care about all that, but unfortunately I do care, specially when in
the preceding days I have edited my third wrong proof of the Littlewood
Paley inequality ( I have just edited, and sent to you, the right one,
but alas, it is much closer to Stein's, and pretty trivial : no more
local times ).
About the Springer story, I bent to Dold : I fought about Getoor's
paper, but accepted the rejection of mine and Yen's provided they
stated explicitly in the table of contents they had been rejected by
Dold and Eckmann because consisting of resumes without proofs. The sha-
meful thing isn't the rejection, but the delay : three months lost in
Springer's offices 1 About next volumes, I don't know what to do, but
I suppose that if I can I'll keep publishing with them. It isn't a matter
of pride, now people know these seminars exist at this place, and
that's important. I was some of nasty with Dold, but he probably didn't
even notice it, he is a good-faith Prussian.
I have been editing the full2ectures on stochastic integrals, and
took many steps in the direction you wished me to, but I think anyhow
you won't be satisfied : it is still very Strasburgian. But chapter III
for instance, just after square integrable m les, is the change of varia-
bles formula with applications, thus showing the difficulties and moti-
vating the local martingales, etc. And I have used as little as I could
of general theory.
It has been extremely nice of you to have written Peters. No one
except Doob and you did.

PS. I was already back in France when Kahane made these comments on
Anyhow, Kahane is great, but isn't my standard as a mathematician or rather

harmonic analysis isn't my standards as a-branch of mathematics. I am more sen-
sitive to your comments, or Doob's or even to comments of people that don't agree
with us, but are still able to understand what's going on like Kac, Kesten,

June 9

Dear Chung :
I am very stupid I Let Pt be the brownian motion on the line, Qt be
the br. mtn. killed at 0, on R+ The function i(x)=x is invariant w.r.
to Pt and odd, so i(x)=x on R+ is invariant by Qt, but now it has become
positive en 7+.
Now extending to En you may see that killed br. mtn. in a half space
always has a positive invariant function. Well, certainly people working
in potential theory know a lot about that since it amounts to the existence
of < points at infinity > in the Martin boundary.
With best regards


TtL (a*) 35.34.0*

Dear Chung :
I was much worried to have exchanged the letters between you and Rao : he must
have understood by my enthusiasm that, alas, he wasn't the right version of the
process. I discovered the truth just last week, and was much disappointed -though
M. Murali Rao is probably a very pleasant visitor, it was K. Murali Rao that I
expected. But anyhow please don't tell the right one : otherwise he might be willing
to come here and I would be unable to. invite him in a reasonable future. And besides
that it would be bad tM.M. to publicize the story ( I didn't even know that ano-
ther Murali Rao did exist ).
Don't worry about Glover's manuscript : it will be forwarded to me in August,
and I'll read it there if I have time otherwise, in September here.
Az4ma and Yor will carry on the seminar volumes after volume XIII. What I'll
do myself is another problem, and far from solved. In 1979-80 I plan to travel
( with my wife, if her health and that 'of my parents allow it ), and quite probably
I'll take the chance to go to China if the possibility is offered again to me.
By the way, I read in the last issue of Scientia Sinica a paper by Yen, which isn't
bad ( nothing fundamental, but nice details, and nice pedagogy ;,it is very cleanly
written ). I boasted a lot about having read a paper in Chinese from the first to
the last line, but none of my colleagues paid attention to it of course, for you
it is a trivial matter.
Az6ma wrote something about the Littlewood crocodile and its Martin boundary
but I don't know the reference, may be it has some relation with that ( very inte-
resting ) problem.
I was amazed that anything in measure theory required such a delicate proof
as your Hahn lemma proof I had always thought measure theory was something that
could easily be reconstructed while driving to the department, just before the
I wish you very pleasant holidays

It is pleasant that you are doing so much French i I am particularly happy to have
French as my maternal language, since it is so difficult to learn ( may be German
is even worse ).

July 12

Dear Chung :

Progressivity : I prefer to withdraw my statement that one should
use [0,t[ instead of [0,t] : I just forgot why J The remark was due to
Walsh, and apparently he had some reason.

Local Martingales : there are two things to remember 1) No
restriction of integrability on X0 2) if X0=0, then obvious definition
by stopping. Point 1) is often forgotten.
I agree that our definition isn't pedagogical, but I don't consider
it a pitfall 3

China : Here let me be careful, so that you may use my name without
written permission by author and publisher

Yan Jia-n quite good, even by european standards. If they
Zheng Wei-An
are able to publish in Annals of Prob. or ZW, why not leave them ? I
agree with you that they are too good for China, i.e. probably not very
useful to the country. This contains an element of instability, specially
in Zheng's case ( he is very young was a kind of prodigy, and is immen-
sely proud though not in an unpleasant way at all ).
He Sheng Wu good, probably at the right level for China
Wang Jia Gang
About Chen Pei-De, he was probably spoiled by the 10-12 years he lost.
I didn't encourage him to study Riesz spaces, I hate them. He certainly
cannot follow Yan to Germany : the Germans are very careful about the qua-
lity of their guests.
E Let me rectify one point : the notes by He-Wang and Zheng were typed
by me, rewritten by me, but the ideas were theirs entirely .

My suggestion is the following : that you should write me a letter con-
taining essentially what you say about this mathematics which
isn't the kind of research everybody even understands, its exceedingly
abstract character, the fact that every probabilist should know at least
the contents of Feller's volume 1 L yourmention of Tortrat seemed quite
funny : Tortrat is 10 times more abstract than any of us and so on.
Then I'll show it to all my chinese students. You are highly respected
among them, and this will have some weight. You may even write it in
Anyhow, you should understand that we essentially agree on this point.
Another rectification : your << may be Yan did some discrete limit
theorems under Rost >>. What Rost does ( infinite particle systems in
Dobrushin's style ) is IMMENSELY more difficult ( and, as I see it, more

abstruce ) than what we do here. In spite of all his eagerness and
courage, Yan.didn't succeed in learning the subject.
Finally, let me conclude that the correct word to qualify what(
do isn't abstract. You know perfectly that what we have going here is
< rectification of names >, and names are used to denote concrete
things. When things settle down, people will notice that everything we
have done here is extremely easy, almost trivial, just like ordinary
calculus but abstract, no.

-"cyTU A\fy ^f-


Sept. 16

Dear Chung :
I really couldn't answer from Paris. So I'm doing it from St Peter,
where I came back yesterday.
Question 1 and la : I just don't understand your proof The U g, gsCK
aren't continuous, how can you apply any Bolzerstrass theorem ? Anyway
you are asserting that any special standard process is a Hunt process.
C'est trop beau pour etre vrai 1 There are counterexamples.
Question 2. Your MOT result is just my I.T20 : the fact that H is a
vector space isn't used in my proof. The fact that it contains 1 is
true in your case, since the space is countable at infinity, therefore
the smallest monotone tass set containing CK contains 1. Uniform closure
isn't necessary, since CK already is an algebra and a lattice. I was
ganz dumm to assume H was a vector space I'll delete that from the
next edition.
Shame on Bourbaki His so called integration book doesn't contains
any MOT

September 25

Dear Chung
1) Yes, I got the card at Vaux, and my parents were glad to receive
something from HangChow, since they had enjoyed their trip there.
But you may understand that the only possible answer would have been
a postcard from Balbronn to Stanford, asking whether you had heard of
such a postcard before. Unfortunately, the Balbronnese haven't postcar-
ded their Dorf, and I had to keep silent until you gave me some other
reason to answer.

2) The proof of 1)=>2) on p.24 seems OK but of course the theorem is
desperately wrong for continuous time local martingales. I don't
understand your sentence on Cauchy variables ... The Tk are the stop-
ping times in the definition of local martingales, which reduce them
to true martingale, so you can't choose them as you'd like to.
Your query about cond.exp. amazes me. I had to explain the same thing
last week to Mr Kailath. There are three cases when one can reasonably
define E[XIF]
when X is integrable ( the trivial case )
when either X or X- is integrable ( the stupid generalization )
when there are sufficiently many AeF such that X is integrable on A
( the good generalization ). Then E[XIF] is just E[XIAIF on A This
is ogten quite helpful, and isn't given in any book.
One interesting remark if my p.24 is true, then it shows how subtle
local martingales are. While a true martingale relative to a family F
is a martingale in sich" w.r. to its natural family, that may be
wrong for a local mart. It may happen for instance that (Xn) is a local
mart. and (2n) isn't, because the first property means that X2n is
integrable on sufficiently many F2n-1 sets, but it may not be integr.
an any F2n-2 set 1

3) Dubins : I paste carefully your letter in my mss, for the next
definitive edition, but I don't want to check it now. I'll test it
in my lectures. I guess my tricky proof was just Dubins's.

4) I am not sensitive about notations I think I am pretty insensitive,
generally speaking ( No, I remember I was pretty sensitive when you
criticized Weil Morando or Dellacherie ).

5) I answered at once : I just came back today from Aumont-Aubrac.
6) Balbronn is close to Molsheim, and Molsheim is certainly on your
card of France-
Sruc 112. 4 ~f

October 22

Dear Chung :
Please send that little note for our seminaire VI. But it would also
be very nice to take your '' semi-expository'' stuff on stationary point
processes The volume is still small.
No time yet to look at K's letter. I don't think the strong Mkv Pr.
can be really harder to prove than the ordinary one.
Is there around in Stanford any good probabilist interested in coming
to Strasbourg next year ? The people there told me to ask. By the way,
it seems I'll get a permanent research position there, a thing as yet
unheard of in France. But it isn't sure yet.
I'll send you some ( half interesting ) things next week from Strasbg.
With best regards and wishes,

A kHL-<

November 2

Dear Chung :
I have little to say about your question. I remember vaguely that if
f < s1+s2 ( f excessive, s1,s2 excessive ) then f=f1+f2 with f1_s1i, f2ps2 :
but I can find no reference for it 1 If I could, I could also check whether
fnts1+s2 implies fnts1 for instance. The fact that if f is a potential then
f and f2 are potentials adds no trouble, since << Motoo's theorem >> is
entirely general, amounting to the Lebesgue derivation theorem along the
sample paths, as Getoor pointed out once,- but a proof for resolvents, without
any use of the process, was given by Mokobodzki. I also think that the decom-
position lemma above is due to Mokobodzki. As you may see, I have become very
ignorant in potential theory, having forgotten even things I once wrote care-
fully Your results with Rao are rather impressive, in my opinion, because
you are proving very precise results with hypotheses one is able to check in
a very concrete way.
This year we are going to study the papers that went last year out of
the Az6ma-Yor workshop in Paris : I think this will keep us busy at least
for six months. Did I tell you that Chou Ching Sung was here ? He hasn't
become a good mathematician, but apparently he is rated very high by his
students as a teacher, and he is also appreciated by his university ( since
the Natl Research Council in Taiwan offered him a year in France ). I consi-
der this is already a big success, since he was so unlucky during his first
years in France with the Dugu6 people, and finally it may also be good for
Taiwan ( he tells me graduates from there with US degrees are rushing out
of the country now ). As a person he is quite pleasant.
Now that the M.M. Rao family is here, I feel unhappy each time I hear
the U.S. 0 is going down 1 Please give all my best regards to K.M.




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November 10

Dear Chung :
I was delighted to hear your disagreeable remark Spitzer's saying merely
illustrates the bewildering fact that people are different from each other, even
good people. To put it in its true context, let me just state that there is no
<< Strasbourg school >> : there is really an << Urbana school >> That is, it
was Doob who invented increasing families of fields, martingales, stopping times,
the strong Markov property and so on, between 1939 and 1949. But of course Doob
isn't a systematic writer, and the writing of his book was anyhow sufficient to
lay the foundations of what came later on. So your statement that Spitzer should
get a sound birching for saying so isn't a fair one : if you replace Strasbg
school by Urbana school in Spitzer's quotation. it becomes obvious that you and me
are on the same side And you may also understand why I am delighted, not hurt,by
I don't like your creation << superpotential >>, because of course it will
suggest the existence of << subpotentials>> and that will add to the confusion.
All things that are called potentials have a vague relationship, and that is enough
to justify the use of a name which, anyhow, doesn't carry enough meaning to lead
into serious confusion. After all, I belong to << potential theory >> and may be
it is the same ( yes, it is ).
I don't remember whether a proof of the theorem on approaching predictable
times by countably valued stopping times was ever published If it was by Dell.
or I, it was certainly a non-algebraic one, that is, using the section theorem.
So mayu Glover's proof as worth publishing. He seems to be very good6 I am slightly
worried about his paper on energy, because I think I found a serious mistake in it,
I wrote Glover, and got no answer. The paper has some interesting things, but I don't
think it has quite the level of the ZW.
You know that Dellacherie is at Princeton ( I remember you mentioned it in
your letter ) but apparently his health isn't good, and he hasn't been able to start
work yet. He wrote me he has at least 20 letters he hasn't answered, and I must say
I wrote him at least 3 times before he wrote back.

November 21

Dear Chung :
I answer both of your letters together.
About the research position : this is impossible still, according to
all predictions. But once it will have occurred, then it will not be
considered impossible any longer Isn't that Zeno's paradox ?
I like the NB, except the last sentence otherwise why are there
infinitely many primes t*Why are there infinitely many integers",
rather 1 And of course this/ question .looks so stupid that it's better
to cut the sentence short at assumptions" -
These are the two last pages of a paper, apparently. May I have the pri-
vilege to see the remainder ? Is that the thing on point processes you
mentioned and which should be translated into elegant French ? I
would do it gladly but that wouldn't be necessary for insertion in
the seminar, and if you/think of inserting it then send it now, since
otherwise it will waste one year.
Also, I haven't quite understood the story about the book : are you
writing it, or is it written already ? Why do you say it is a MODEST
book : because it is short, or because it proves deep results with
little general theory instead of japaneseeing like other books on
boundaries ( including mine ) ? Please excuse me for asking too many
direct questions : I'm afraid I may have lost the little subtlety I eIeN
had a force de manger des saucisses et des choux.
Now about your second letter :
If you want to be absolutely correct then write F for Ft
and X for limt X and everything will become clear ( and a little
heavy ). oo must be a point like the other ones, that is, a disconti -
nuity may.appear there, though in most cases we don't want it, and
then make the convention that X is taken to be X and F =F .
00 O =-00o =co -
About the other notation : I never use a notation like F'+ or
(_)+ My statement is much more simpler than that : the family FtL is
right continuous, so there is no need to put a + anywhere May be it
escaped me in a proof. Anyway, I agree with you.
By the way, I must return your letter, unfortunately : I have no
chance to do a xerox before several days, and don't want to delay my
answer that long.
Now about your paper. First, let me thank you for it : I read it
somewhat superficially, and like it much. It is true that proofs are
much shorter than the usual ones, and it's beautiful to get the results
for r.v. which aren't stopping times, but only birth times for the
process. HOWEVER you have an obvious misprint on p.2, line -7 (Xt-
instead of XT_ ), and slightly disturbing small mistakes The first

one arises on p.5, corollary 1. Since you don't assume that Tn,T are
optional, the FT aren't included in F unless T is F -meas.. Thus
the proof falls &own unless you add either that Tn are optional or
that they are FT-measurable.
The more disturbing small mistake occurs on p.3, line 15 : it is
sufficient to show that each Xt belongs... Then we discover with inte-
rest for the first time that you are working with the natural family
of (Xt), and not with arbitrary families to which Xt is adapted. More
precisely, you must ttate on p.1 that Ft=a(Xs,s sets of measure 0. OR may be it's sufficient to assume that Foo is
a(Xs, set+) indeed that would be satisfactory, because more choice
would be left for F but then may be some trouble would occur in
prop.3, p.6, line -5.
These small things would be easy to correct for me without asking
you again, were it not that my typewriter types much larger than yours.
So please send me the corrected lines to paste at the proper places,
with the same typewriter. We have some time left : Cartier hasn't
sent me his exposes yet.
You may be interested in knowing that ( for arbitrary T )
YeF <-<> there is some predictable process (Y )o
YeFT+ <=> ..... progressive ... (ACHTUNG not well-meas. unless
T is a stopping time 1 )

With best regards,

M0 e QA"

December 13

Dear Chung :
I just got last week your letter of October 21, delayed because
of the french postal strike. I was delighted at the idea of getting
the cassette of chinese poetry, and I have started reading le pavilion
du vieillard ivre ( I have both the chinexe and a french translation ),
but very very slowly my chinese has suffered much of our family situation
( very serious psychological trouble with my elder daughter (15)), and has
been reduced to some spare time on Sundays. Mathematics still has priority
over chinese... The volume with Dellacherie is now ended, except for the
preface ( it goes in a sinusoidal way between Dell. and I, he taking out
my insolent statements and I adding fresh aggressive ones ). We have de-
cided to dedicate it to Doob ( Knight told us 1975 was his 65-th birthday,
and there aduld have been some ceremony, but he didn't want ). If you
disapprove, please write at once, but in any case keep the secret. We don't
want him to know before the book is printed, i.e. before it's too late to
cancel it.
I'd like to' see your "rigorous L6vy" may be I can understand it ?
Yan Kia-An didn't write you. Probably it isn't anything connected with
politics, but rather the fact thathe is really modest and impressed at the
idea of writing to such a well-known mathematician. So my suggestion is
to forget for a moment that you were born a chinese, to consider yourself
as an american, and to write him directly in spite of the fact you are
older than him.
Maisonneuve has found a position at Grenoble, as a mathematician but
not at the scientific university. His address : IMSS ( Informatique et
Math6matique des Sciences Sociales ), BP53, 38041 GRENOBLE.
I don't know what I'll do this next year. With the family problems
I'll be happy if both my wife and I are still alive the cassette will
help me but not her to live 1 Anyhow, I very much hope to see you in Europe.
With many wishes to you and your family for a Happy New Year 1975 L

1. No personal attack, but still aggressive, and he deletes them because
he thinks people may misunderstand them as personal. For instance, he cut
out sentences he thought could have offended Dynkin Of course he's right.
I haven't forgotten the Kesten story.

November 6,1963

Dear Mr. Chung :
I have a simple result akin to your problem about the right
continuity of the family A> Ft a right continuous family of --fields), Let Q.2 1be two sets,

with two right continuous families of ar-fields Ft, Gt and two
probability measures P and .. Take now the product space O !
with the measure PxQ and set :

Ht = completed l -field of F xG with respect to PxQ
Then the family Ht is right continuous.
Take indeed a t and a sequence tn which decreases to t For
every integrable r.v. Y on the product space, set Yt- E[Y!Ht],

Ytn = E[Y|Ht1], and consider the linear space of those Y such that
Ytn-> Yt a.s. as n->oo This space is dense in L1 because of
Doob's theorem : Awm-cfi hA
e.P[ sup IYtn > e ] E[ i ] as
On the other hand, it contains all r.v.'s Y(o).Y'(a'), with Y and
Y' in the L1 spaces of P and Q respectively, because of the right
continuity of the families (Fj) and (Gt). It thus contains the

whole of L.

Now, if the family was not right continuous, there would be an
event As Ht. A Ht. Taking Y=IA, Ytn would be Y for every n,
and Yt# Y ; hence we would have a contradiction.
Very truly yours :

52 rue VMonge Paris 5, France.

March 3, 1964

Dear Mr. Chung :

I have included the two results in my book Indeed, I found your proof
difficult to understand, and I had to rewrite it for myself. The main
difficulty for me cafe from formula 52 :
{(a(t,w),w) : a(t,w) has no meaning at all in my opinion : in {x:P(x)J, you cannot logically
replace x by a composite expression : x is just a letter denoting the
generic element of a set. I thought of sending you my sW ( version of
your) proof, but finally this is a mere question of taste, and you cer-
tainly like and understand your own proof better. In any case, many tx
thanks for all that I had been unable to prove this theorem alone.
As for the overlapping f our works, I have said the following
in my book ( approximate translation ) :'most of the results concerning
measurable processes and stopping times are very elementary : many of
themcan be found in Dynkin's books or as lemmas in papers concerning
markov processes. The systematic study of these notions, undertaken
independently by Chung and Doob and by us ( i.e. by me, as an Author 1)
x led to the discovery of several very useful results. The work of
Chung and Doob [1] goes much farther than the present one G a
There are unfortunately no exercises in my book. 1 found enough
to write the indexes, bibliography... but not for exercises. In fact
han I have also stopped before starting markov.processes : they will
be left for another volume. I have included in *A the results of
my paper about stopping times in progressively measurable sets ( that of
the potential theory seminar). I discovered xiR I had said that
something was obvious in that paper, the proof of which takes 4 pages
in my book.
Genevieve and I send our best wishes to you Mrs Chung, and
your family .

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April 9,. 1967
]har Mr Chung :
Here ( I hope ) is the answer to your questions. Some of them, as
you remarked., are settled by the fact that 4A so the only questions
remaining are those concerning left hand limits.
Hunt processes : they exist in' far every t, and belong to) E for
t<, (af course we may have X_ s) .
standard processes : left limits exist in w E' for every t X_ may not exist, and in this case A) ,, and in particular
far t<4 They are in E for t< .
Now to the proofs : it is true I have not stated the fact that left limits
are in E for t as for the proof, this is a particular case of a theorem about hitting
times given (, for Hunt processes ); at n7 of chapter XV. This is also a
particular- case of the result on standard. processes.
In that last result ( T23), the statement is correct, and apparently
my mention of 1) at the top of p.100 refers to an earlier organization,
where 1) was probably the existence- of left limits in E', while 2) was
the fact XtaE for t( But I simply forgot somehow to prove 2 !: this is
unforgivable Here is a proof. I take the existence of left limits in E'
for t greater than 0, and get a contradiction. $SC<}
Let S inf t : t Xgs or SB Indeed], if X (ca)#h there exists ( since S(C) doesn't
belong to the set t :...1) a decreasing sequence tn of elements of this
set, strictly decreasing, converging to- S(') Then XS(W)" lim Xt (W)=
lim Xt () ( we may assume the tn are-XA)= thus S(4) >()n

Set T= inf it : Xt_-=3 00 by our ( con-
tradictory) hypothesis and Ts S onjTirzo, therefore XT from the discus-
sion above, while Xs# for all s< t Let. Vn be s-eence of closed (in
E') neighborhoods of 3 such thatI CVn then XT_=6 implies there is
a)0 such that Xs ,Vn for se(T-a,T( and therefore TV < T on T Let T'l T be the limit of TVn ,, we have T'<, onT l(oo n and therefore
by quasi left continuity XT,71 lim XT a.s., implying T> T' the
wished for contradiction.

Anyway, I'm afraid you'11 be somewhat disappointed in your hopes of
finding in this chapter anything really useful to boundary theory : you
need probably less general things, and deeper. I undertook that only to
fight ( mildly) Dynkin's ideas, and to prove that all the properties he
has ascribed to his complicated "processes" were in fact properties of
the semi-group, and that, if one could prove them for the canonical rea-
lization they were true for any process ( a la Doob i.e. without Px" 9t
and so on).
There was a Canadian student here ,. Gaston Giroux, that *gp interested
r boundary theory ( I say : was, because he is leaving foar Canada this
week it6emiftwwill have a child soon ). So his plans for next year are
somewhat dependent upon yours : if you plan to stay here 3 months, or are
in uncertainty about the time you'll stay, he will probably come alone for
the first three- months, while if you are sure to stay longer he might try
to come with his wife and child. So please let me now as soon as you are
sure of Eam either eventuality ( just let me now : I am not putting any
pressure upon you because of Girouxr- though of course I would prefer the
longest possible stay !! )

a Because a I advised him to come alone for 3 months in both cases : he
came with his wife last October and she said she didn't like France and
returned to Canada after 10 days ( of course, she was also beginning to
feel she was pregnant) She will probably like France no better next 9ear?

October 26, 1968

Dear Mr Chung :

First of all one thousand apologies for not.having mentioned your book.
I received 'it in September. It seems to be quite an exciting book for stu-
dents, with a lot of mathematics and abstraction kept to a minimum but
I am unable to do any detailed comments now, since I didn't look at the
proofs at all. I think our librarian must have ordered several of them now.

Certainly the wording of my letter to Dekker wasn't the right one ( by
the way I wouldn't have written to them, hadn't they asked me ). The
kind of advantage the University would have obtained from a Lecture Notes
publication could better be stated thus :

1) I would like to have one copy for aach probabilist member of our
department : i.e. Bretagnolle, Dacunha-Castelle, Fernique, Foata, Fuchs,
Cartier, Della cherie, (Meyer).

2) Also, to send a copy to former probabilist visitors : Spitzer, Joffe,
(Chung 1).

3) Also, to give one copy to : Morando, Doleans-Dade, and Weil.

4) Finally, to have a small number of copies (5 or 6) fer unspecified

We reach thus the 20 copies I mentioned to Dekker ( except that the
inclusion of Chung ( and Meyer) above was a trick ), but obviously only
4) requires a formal agreement with the publisher : the other copies can
be just included on the mailing list.
Another thing that the University would have got from a Lecture Notes
publication is prestige As you know, we are a small university, and our
neighbors are very big ( a recent paper from the university of Paris said :
il faut bien expliquer aux 4tudiants que la recherche math4matique est tres
difficile, et qu'ils seront en competition non seulement avec les math4ma-
ticiens PARISIENS, mais aussi avec les mathematiciens ETRANGERS ). May be
something about the excellency of our choucroute can be included in the
To end this matter, I want to tease you a little ( not seriously ) : you
say 1I fortunately, I signed nothing... I only regret that I even discussed
the matter with them until the MS was ready... I do not wish to subsidize
them to this extent" Now you certainly are right that they are living
at the expenses of mathematicians, etc, and that the Lecture Notes are not
the proper place to publish a manuscript that has been well polished like
yours, but you are too passionate ( in conformity to your legend )! You
have just forgotten that the idea of publishing Prof. Chung's lectures in
the Springer collection originated with... Prof.Chung himself. I perfectly
remember when we first talked about that in the department lobby : you
were telling me that you were preparing your lectures very carefully,
writing everything, and that there would be a number of improvements, that
it would be nice if some student could take notes for an informal publi-
cation may be even in your Lecture Notes collection". So it is entirely
natural that you talked with Springer about that, and you have nothing to
The section theorem that my pupil trivialized ( as you may gather from
1) above, he is no longer my pupil : he has been promoted ) is the section
theorem : given an optional set, you can find stopping times ( sorry :

optional fandom variables ) running into it. Now that it has a decent
proof, thb theorem will probably become quite classic 1
I am teaching this year my new boundary stuff ( essentially the old
Kunita-Watanabe one of last year, with a great deal more on coprocesses).
You'll get it as soon as it appears ( within one or 2 months). I have
taken the risk of publishing -the end before it was taught -thanks to the
fact that about 3/4 of 11 x agt the lectures had been taught last
year, and very-carefully corrected ( with your help). Also, Bretagnolle
Castelle are lecturing on'Ornstein's theorems, and we plan to study also
a new paper by Stroock-Varadhan on diffusions, and Kestdn's work ( he
-sent me a -copy o-f his MS on your conjecture, not yet on his new results).
Very truly yours, -

September 2? 1971

Dear Chung :
Let X be a right continuous, strong Markov process, with state space
1) if X is Hunt, and g is regular, then left limits do exist up
to oo and (goXt)- goXt_ ,
2) if X is special standard g is regular, then left limits exist
up to C and (goXt) = goXt_ up to C.
3) If X is Ray, g is regular, then left limits exist up to c and
( as 1)).
These three theorems are really the wame or rather 1) and 2) are
particular cases of 3), because : in 1) and 2), one has left limits
in the topology of E, and.they happen to be the same as the left
limits in the topology of the Ray compactification ( up to C in case
2 ).
Now I.must say I can't follow your.proofs because I don't see
clearly which is your aim : let me try You would like to prove
without Ray's resolvents that for a strong MP X and terminal time T,
X continuous at T => T accessible ? But you allow yourself to use
the full general theory of processes ? On the other hand, this isn't
exactly what I had proved f.or stopping times : I had a little more,
namely that, if XT_ didn't exist or XT =X then T was accessible,
but I agree the extension is trivial, since {Xt- doesn't exist} is
a countable union of predictable graphs.
Let h be the 1-excessive function E'[e ] : that is 1-0 in your
notation. I really don't see why the continuity of X. at T should
have anything to do with the regularity of h, which should mean that
if SnTS then Tosn -:>ToGQ a.s.. So there is no point about using the
above theorem. It might be true, however, that ( as you say ) hoXT_
=(hoX )T_ a.s. : my as you say" means that I'm not quite satisfied
with the wording : since the process has no left limits, XT_ is mea-
ningful just because X T=X and then why not write just XT ? But
unfortunately I find two spots in your proof. The first one isn't
bad : you assert that A is a CAF because T is terminal, but it is
kein CAF, nor even a 1-CAF. The second one is more troublesome :
H is a countable union of graphs of predictimes, the graph of S is
contained in H but that implies just S is acc, not pred.

My final comment is that it looks very complicated to simplify things. It
might even be against la line du parti to try to get rid of Ray : it's so
nice to have a simple theory which contains as particular cases Hunt, stan-
dard, standard special, strong Markov processes." First climb the mountain,
then look at the clouds under your feet" says the ( home made ) chinese
With best regards

A kig

February 1, 1972

Dear Chung :
1) About7-MCT : An obvious answer to your question : if H contains C ,
H is monotone ( not vector) C is vector, A-stable, and
feC =:> fAl eH
there exist f eC f tl
n = n
Then g'= jf+a feC, ae I is vector and A-stable, contained in H and
then H contains bC This was my small proof from last August (?).
However, there cannot be a universal MOT. For instance, on a polish
space, any monotone set of functions that contains the positive l.s.c.
functions contains the positive Borel well, it also contains the con-
tinuous ones, so that some vector methods might do, but from an abstract
point of view., this .corresponds to the fact that any monotone class con-
taining the open sets contains the Borel, because the complement of an
open set is a G,, and there are no vector ideas involved.See my MOT for
sets in the French edition of my book ( I changed it between the editions)
So my advice for MOT is to give one good theorem for sets, one good
and simple one for functions, and a method, may be ;some other ones for
the exercises for instance, cay you prove by MOT methods that any Borel
3 function is close in L to a C function ? that it lies between a lsc and
an K
usc function whose m=xmNxN integrals are very close ? etc.

o 2) About Dellacherie. I found you were very naive (, I hope I am not offen-
sive 1 ) to believe that Dellacherie read Getoor-Rao, or reads the papers
of his master, and didn't read you He simply doesn't read anything ,
like 990/o of us ( in the 10/o I include Neveu, and you ). It's so terrible
tiring to read even one page in the seminar 1 To follow somebody else
> without knowing where it will lead you 1 He mentioned Getoor-Rao because
- you mentioned them in the last four lines of your page, but he didn't
3 read them. He started thinking about the theorem and found he understood
Sit and it was trivial. So he was happy and didn't look any further.
_3 'Thus your page has been useful, after all.
As an illustration, let me recall you that I found it easier to write
Knight-Pittenger than to read it...

3) About the position : I GOT IT. But I don't accept your suggestion that
it may do any good, or that I ever did any harm, to MPajdP. First of all,
I .suspect potential theory is immortal. It's 100 years now since Gauss
first "exhausted" it and then Poincar6, Brelot, Choquet, Doob, Hunt,
Mokobodzki,(,rostma, and many more... I'd even say something more about
it : its devilish I hate it, and I am potentializing all the time. Now
that MP has become associated to it, it is immortal too, and Karlin, Kac
and Kesten have nothing-to do against it. Anyway, I baW your putting

Kesten with the other two, just because ha begins with a K/: his paper on
processes with independent increments was just as much MPandP as, say,
Knight-Pit tinger.
Moreover, both you ( when you attack generalization" ) and me funda-
mentally agree with the critics again the present trend of the theory .
The only point of disagreement, and of stupidity on their side, is that
they ( i.e., Karlin, neither K. nor K. have anything to do with it ) do
not want to hire Walsh, and that/because Walsh is outstanding, at least
.when he gets warmed by the surroundings.

4) I have got a very interesting paper by Joseph Horowitz ( he, and myself,
are counterexamples to your theory that Jews are against MPandP ; Weil
too Dynkin too, I heard, but may be you'd not accept him among the right
MPandPers : so some hypothesis must be missing ). It concerns flows, and
Palm measures. I have started thinking about it, too, and I discovered
that everything about fPalm measures, etc (/Ryll-Nardzewski, may be quite
a bit of OCramer ;,,?) are contained in the old, old paper of Ambrose and
Kakutani (Annals 1941 ) of flows under a function. But Horowitz goes
farther than the streams of calls, and quite a bit of flow theory gets
closerto MP. I'll lecture on flows, so I'd be very much interested in
your semi expository material.

5) About 3), I forgot to say that Strasbourg is getting quite suspicious
of my school, just as Karlin may be suspicious of Walsh. But now that
I succeeded in smuggling Weil and Dellacherie into the place, and myself
out of it, I'm happy, and I don't fear anything. But the colleagues refu-
sed to hire Smythe -for one year, and hired instead Van Moerbecke ( a pupil
of McKean and Kac ) believing he was a statistician, while he is at least
partially interested in MP I'll corrupt him i

6) About collaboration : theoretically, with joy Practically, since I
am here ( 2 years ) I have started doing 36 things, and completed nothing.
Now I must finish at least the elementary part of the book on martingales
and give it to the L.Notes before the Summer ( the difficult part I'll do
with Dellacherie as co-author ). So my proposal is to wait just till
Easter : just now, I don't feel free However, may be the subject will
be overwhelming ? Let's try .

vA fcl

67-STRASBOURG.L October 30 1972
TL. (sa) 35.34.02

Dear Chung :
I just got yesterday your letter with the corrections for which one thousand
thanks but no comments yet ( since I didn't check them yet on the book ). Only
comment : I will NOT pbad guilty about my supposed crime : please look carefully
at the statement. I agree that f finite is a crime, but H f defined ) is almost everywhere finite is perfectly correct. The fact that
E[fIF) f.dP is sigmafinite ( f positive, of course ) was obvious for
me, and I was puzzled to know it could be non-obvious for anybody ( for instance
t6 Kailath ).
About Kailath : I just don't know him, I talked with him about 1 hour in Freibg
I am now writing an expos on his"innovations" : the subject is very difficult
and interesting and he ( T.K.) has had at least one very beautiful idea. I wish
there were many mathematicians like him, that is working on interesting problems,
not on the problems they can solve.
Maisonneuve finishes his thesis this year his work is quite good. He doesn't
know semiclassical potentials, but if he is warned sufficiently in advance he
can learn. I can commend him warmly : he is still A'lightly immature, but original
and hard working We ( he and me ) would like to send him to the US next year,
so why not hire him ?
With all my regards,


^Thjo C'KUJ
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October 16, 1974





t 0

E' =
ar 5

Dear Chung :
I asked Mr Yan Kia An to send you the information you wrote me.
May be he already wrote you, but may be he won't : he is very careful
never to touch any non mathematical subject, and he is also somewhat
shy. But he is very pleasant.
Knight has a beautiful new paper not that one on the germ fields,
but one on the general theory of processes which contains a rather
amazing result. He wouldn't consider it that way, but I would classify
it with Schwartz's paper on the existence of kernels yielding the well-
measurable or predictable projections but without, of court the
overaxiomatisation and L4 al I may state his result :
consider a state space (0,F) (nice) with a nice increasing family
of sigmafields Ft Then you can build predictors relative to Ft+
which are nice kernels, depending measurably on the probability law on
0, and which can be interpreted on a suitable space as deriving from
anomogeneous strong Markov process if 0 has a shift operator Ot. I am
very vague here, but in my opinion this is the main point, and fairly
non trivial.
Dellacherie and I also have a new section-projection theorem, which
is true on any probability space with any family of sigmafields Ft,
that is, neither right continuous nor complete. Take the most elementary
case : that of a martingale. There is one and only one version of the
martingale E[YIFt] neither right nor left continuous which is
optional with respect to Ft ( that is, measurable on the sigmaf. gener.
by all rightcontinuouswithleftlimsadaptedprocesses, no completion at
all, that is for instance Yt will be just plainly Ft-meas ) and satisfies
the stopping theorem relative to all stopping times. Of course the only
one is up to undistinguishability (20 letters ). Nothing is really
deep here, but we had never thought about it, and even Mr Yan couldn't
prove it at once. on stochastic processes
Chapter I/of probability and potentiels will be very Chungian :
it contains right separability, essential topology, and the roman theoren
on approximating predictable stopping times by dtages ( with your proof).
Of course it is also very Doobian, even rescuing an old theorem in
Doob's book on the approximation of integrals by random Riemann sums
which seemed completely useless, but now gets a meaning. Writing this
chapter was a delight for Dell. and I but it may be somewhat too
obscure for the non initiate. I hope you'll like it.

I gathered from your letter that my self-taught chinese hadn't
seemed ridiculous to you, and that made me happier than a favourable
mathematical comment.

~^ H

-- D67 -STRASBOURG. Lx March 17, 1975
TL.. (se) S9.34.0a

Concerns t Lecture Notes in Mathematics
S3minaire de Probabilit6s IX

Dear Dr Minio :
I was glad to get the first news of our manuscript, sent to Sprin-
ger last December.
I can consent to the withdrawal of my manuscript, and to that of
Yen, though I feel there are good reasons ( usefulness to mathemati-
Z cians working in this field I ) in their favour. The clear and concise
account of the strikingly new results of Yen on an old subject, while
-the complete publication will take place only in Peking, and the care
E with which he wrote it and I revised it, for instance. But he's my stu-
dent here, and won't get offended since I am treated like him.
On the other hand, there can be no discussion about Getoor's paper
on kernels. You should never believe Prof. Getoor when he says his pa-
pers contain nothing new : he's just modest I confess this may be
misleading while so many young people claim they turned an old piece
of furniture into a new one just by sticking some of their chewing gum
to it. I asked myself for this paper and it is an honour for me and
for you that it is published in our volume. If you want to reject it
Please return the whole manuscript at once.
If my judgment on this paper isn't accepted, anyhow, I don't see
how I could continue to act as referee in any Springer journal. I hope
you don't mind if I circulate this letter among those of the editors of
the Zeitschrift fur Wahrscheinlichkeitstheorie I know personally -
S and some friends : one of the aims of our seminar consists in making
probabilists merry, and the rejecting of the Getoor paper is worth a
^ good laugh among specialists. May be some reactions wil. be contributed
to the Mathematical Intelligencer.

By the way, I cannot believe this decision may come from Profs
Dold or Eckmann : they know much better. Your administration must have
'interpreted" them.
I cannot understand, on the other hand, your policy of rejecting
about 25 pages at random among nearly 600, after insisting so much
last year that is was so hard to paginate the volume and type the table
of contents. You'll have now to do all the work again after we spent
days doing it. That's Operations research '
Welcoming you as a new colleague, and with many regards to the
older ones,
Very truly yours I

SPRINGER-VERLAG Berlin Heidelberg New York
MOnchen London New Delhi
Paris Rio de Janeiro Sydney Tokyo

69 Heidelberg 1
Postfach 105280
Professor P.A. Meyer Neuenhelmer LandstraBe 28-30
University de Strasbourg Telex: 04-61723
Departement de Mathematique Telegram address: Springerbuch
Tel.: (0 62 21) 48 71
Rue Rene Descartes Extension (06221) 4872.50..
F 67 Strasbourg your ref.
your letter of
our ref. RM/HW 2141
date March 13, 1975

ref.: Lecture Notes in Mathematics
S6minaire de Probabilite IX

Dear Professor Meyer,

Thank you very much for your letter of February 11 with the in-
formation that the book on potential theory will be published by

We apologize for not writing you sooner about the state of the
seminar manuscript "Seminaire de Probabilite IX". The editors
have reviewed the manuscript and felt that some of the contri-
butions do not satisfy the Lecture Notes in Mathematics criteria.
In particular Yen's paper (p.p. 444-450) and your paper (p.p.
502-514) resume without proofs. Also, Getoor's paper is, as he
says in his introduction, purely expository. These articles
thus conflict with point 3 of the enclosed list of rules. The
editors feel, therefore, that these contributions should not be
included. Please could you decide whether you would like your
remarks following Getoor's paper (p.p. 472-473)'to be included

After these changes the manuscript will go straight into pro-

Once again we would like to apologize for the delay.

Yours sincerely,

Roberto Minio
P.S.: I am a new colleague in the Mathematics Editorial

Postacheck Berin-West 1730-106 Bankkontn Berliner Disconto Bank AG, Berlin 021/6176 Berlner Handeltgeselechaft -
Deutsche Bank AG. Heidelberg 04/62275 Frankfurter Bank, BeSain 14257
Beriner Bank AG, Berin 99/07572.500 Chemical Bank New York ured Frankfurt

April 14, 1975

Dear Chung :
It is very strange that I like receiving your letters in spite of
the fantastic amount of unpleasant things you tell me. About Minio, of
course he didn't tease me of his own decision, but the origin doesn't
lie with higher people, and in probability" as you say. Alas, you are
worldly enough, but you have spent too much of your time dining and
gossiping with Fields medals, you have forgotten that in most countries
{higher people Inin probabilityl=0. The rejection comes from Dold, the
Springer people are sick with our seminars and want to get rid of them
in some unofficial way did I tell you that last year the manuscript
was "lost" for two months ? the reasons may be personal, but I suspec-
that rather, to these German algebraists or differential geometers, I
am, you are too "statisticians". Remember your letters concerning
Kesten's sensitiveness, well, being a probabilist9is somewhat similar to
being a Jew, you may be admitted among higher people, provided you never
forget your origin and behave modestly. I'll send you some day my answer
to Dold : I consider it a ood answer, but your letter was so nasty that
I may be nasty too!n, `UC I.j/iou f-
About the fact that I just spend my life rewriting other people's
paper you may use this letter as an official and public confirmation
of it ( as far as wRat I may or may not do can interest anybody ). Even
in my thesis I"rewrote"'what the Russians had done, in the stochastic
integrals what Kunita-Watanabe had done, etc. It seems that my only
original paper was the one about the construction of semi-groups in
Brelot's theory. For this one I didn't depend on. other people.
I am glad you didn't read my Littlewood-Paley proof : there is a
serious probabilistic gap in it : formula (21) is wrong, conditioning
repla(i-ag the Cauc y pr cess by a different one. But you sho'ild still
have a glance at the art concerning local times. The subject is fascina-
ting, related to your results on maxima ( I am waiting eagerly for the
complete version ). I will rewrite the whole stuff when I find time for
it also my lectures on stoch.integrals, I entirely agree with you.
But don't be too severe : my personal situation has been changing, but
not improving, and I haven't been doing any mathematics now since before
It was really nice of you to have written Springer, and to say you
hope the seminars will continue, even more encouraging, that you suggest
inserting in them the Skorohod note and the Durrett corrections to Ito
McKean. I don't think in the present situation with Springer we should
make any change to this volume like inserting the Skorohod note : it
would in some way justify their delaying it so much. But anyhow if the
volumes continue you must feel free to insert in the volumes anything
you want.
With thanks again, and regards

P,- K.l I

ak -ai&
ALL 2Autc A

ca ot

March 9, 1978

Dear Meyer:

Mainly I am writing you since you told me that you had helped
in the volume on integration by Bourbaki. For the last two years
I taught a course to "undergraduates" on Lebesgue theory. The first
time I used the book by Kolmogorov-Fomin, which does measure first
and defines the integral by use of simple functions. Now I am teach-
ing from a book by Kenneth Hoffman (Analysis in Euclidean space)* which
uses the Bourbaki approach: first integration a s1 Riemann of 0 func-
tions, bhen define the Lebesgue integral by L -co~letion. Thus f is
integraT ifthere is a sequence of 0 functions which converges in norm
and also almost everywhere. To prove monotone convergence theorem, one
must then use a form of Lusin+Bgorov in order to get a subsequence to
converge a. e. I assume this is the standard way the theory is taught
in Strasbourg, Paris a. e. But except in the book by Dieudonne which is
after all equivalent to Bourbaki I cannot find any text which uses the
latter approach. My colleague De Leeuvw told me that he had never gone
through it. It is to mY mind (and DeL's) very ntntuuitive and puts too
many difficulties at the beginning of a technical and boring kind. Of
course we know general Nikolas is above such considerations but, jokes
apart I want now to ask you what the advantages of the latter approach
are. ?? Of coursethe functional aspect of the integral is very useful
and theorems such as Lusin's also, but these can be proved without more
difficulty after the integral is defined through simple functions. In-
deed., if one does not assume a primitive form of Riemann integration
things are less cluttered and more fresh, because many rep Mons are
avoided. Further*, ven Bourbaki now realizes that measures of sets are
useful on their ownsbui to get these from C (through Tietze/Urysohn)
is so contrived compared to the set approach. In facts Hoffman defines
measurabilty after integrability by use of a. e. convergence of 0 func-
tions, and still takes a long time to get a countably additive measure.
[Only the concept of a set of ouetr measure 0 is used (and necessary)
for his definition of Lebesgue integral.] .....
I know when Bourbaki's original volume came out (when Feller* Kac
and I were in Cornel., around 1948), there was much raising of eyebrows.
The historical approacK from Borel [whom Bourbaki evidently did not wor-
ship], Lebesgues and then F. Riesz, seems also didactically the correct
line. More so when we teach undergraduates and wish to impart to them
something NEW (not warmed-over very tough Rxmax extension of Riemann
integration they learned in Cycle 1!). If one does not worry about
4ooeyA theorems such as LusinJs the usual approach is so simple and results
like the monotone convergence so crystal clear by increasing sequence
of simple fAnctions'as we all do in probability. So why did Bourbaki
choose its way? I have a linS! gp t for him and that is why I
am aj~g nu~% un iwtir. I only did a little very old-fashioned
limit theorems. Here is an example: if Xn is a seqt of indep. ident.
distr. random variables with mean l1 and eefIXlog IX1I) < oo then
cfAb- E(X( -1)/Sn converges a. as. Sn K
Also I get a domination for n/Snbut d
n but d don't know any use for it.

TEL. (88) 35.34.02

Dear Chung :
I was happy to receive a letter from you I entirely agree with you :
.Bourbaki's approach to integration is ridiculous. My collaboration with him
concerned only the last chapter, and of course the other chapters had to be
.taken for granted. Bourbaki himself agrees that if the whole thing had to
be written again, then he would do it quite differently. On the other hand
rewriting it would be a big work, and essentially uninteresting since there
are so many good textbooks on integration now, and Bourbaki has better to do.
The chapters stand now as a very complete text on a particular theory of in-
tegration, a good reference, and something to avoid absolutely in teaching.
The story I heard is the following : Bourbaki had started with abstract
measure theory, and written a lot on it, something so enormous that it explo-
ded, and so they decided to rewrite it in a <> set-up. This is very
understandable : that first version was prepared in the forties, before, say,
Prohorov's paper on metric spaces, etc., and it wasn't clear at all which
was the good theory.
I never got the paper by Balkema. Why ? He may still send it to me for
volume XIII. Volume XII is very thick (800 pages ), and may be not so good as
some of the preceding ones. Obviously the system needs a change.
The Chinese invited me to spend 3 weeks in August-September, but I don't
think I'll go. The problems with my daughter aren't quite solved yet ( and
it isn't sure they will ever be sold ).

June 2, 1978

Dear Chung :
I was glad to get a letter from you. I don't think it interesting to run into
Dieudonne : this old battlefield of measure theory has become quiet now, it is
useless to pick up these deadly quarrels of the fifties. May be even Dieudonne will
acknowledge Bourbaki was wrong and after all, was he wrong ? It is nice that Bki
was written, so that nobody will try to follow this path again.
I am slightly skeptical about Schwartz's conditional probabilities, as you
are, but on the other hand these ways of stating results must be explored : if
time doesn't belong to R, but to R 2, as it has become fashionable now, stopping
times are missing, and may be Schwartz's approach does work. Besides that, Schwartz
has become a true probabilist, not just an analyst that has turned on to probability
because it is so much easier than mathematics : he stimulated Yor into proving a
nice set of martingale inequalities and if this isn't probability...
It is very nice that moderately Mkv processes get studied at laskl last !
I'll be glad to referee the Mss. Unfortunately, I don't know anything about your
invariant function problem the obvious person to ask about it is Doob I
believe there are no invariant functions for killed semigroups, but I have no reason
for it... What about the domain between two rays from 0 in 2 ?
I don't like to talk about that Chinese story : I am not going, as usual for
family reasons. I was very sorry not to go, and they took it rather badly at
least I guessed it from the fact that they wrote me three extremely polite letters
to have me come to their embassy, but didn't even care to answer the letter in
which I told them I couldn't go.

1No, it is not true I was very sorry not to go : I thought for a few days I would
go, and it made me terribly uneasy : they have such an extreme consideration for
science that it would have been UNBEARABLE for me. Imagine that when I went out of
the embassy I was between two Attach6s Culturels, each one respectfully carrying
one of my bags. They also were very much surprised that I took a bus like an ordinary
mortal, instead of sitting in a taxi as everybody does in a capitalist country.

Oct. 12, 1979
Dear Meyers

Under separate cover I am sending you a MS by me and Rao. This
contains results three years old and is written up because we have
not been able to do better. Indeed we don't know what to do next.
For this reason I urge you to look at it and send your comments.
I should tell you that our idea is to apply the results tb a large
"ball" with regular boundary, and to derive all cC'f&sical results
in the symmetric case by the modification, namely extending them
to the nonsymmetric case. But we don't know how to show that the
function of modification 6 is "nice" enough to get those results.
Perhaps we don't need to know this? Anyway the problems are wide open
and I just hope some brilliant idea will occur to someone!

On the other hands I have some to me very staisfying results abput
a potential theory with the Kac functional and many interesting con-
sequences for the Schroding,;er equation. Here is a key result. Let
G be a bounded domain (con:'. i/d X be the n-dimen Brownian motion:
UG(x) = E^ exp (f q(X(t))dt. f(X(T(G)) ), T(r)=(stk tVhe

where q is bounded (Not of constant sign!) f is sb iat~ive Then
(a) if uG is not iden-cically m in G, then it is bounded in G
\jet without wahtever condition on IG,
C.A. p-y -Q / (b) as x tends to a bay point z regular in the usual sense (i.e.
VA a reg for CG), then u tends to f wherever f iF continuous

(c) if q is Lolder continuous the' u satisfies the Sc34odinger
eq. 4h+
e+ q) u = 0 in G
,.dCV --) (A) (more than (s)): under the hyp in (a), there is an open H
containing the closure of G such that u is bounded in the
closure of HI etc. A transfinite induction should yield a
minimal domrin in which the associated v is identically oo.
This is related to the least eigenvalue of the eq.
Surprisingly the boundednees of u in G under (a) is not easy as
Doob learned to his chagrin [he reR ains the only expertt" who tried/]
Therefore I am submittingtto French experts to try too. In view of (c)s
the result implies that the axiomatic potential theory a la Brelot ap-
plies at least to the Schrodirner equation. [It is easy to deduce other
results such as the corresponding Poisson eq.] I believe that some
French analyst Perhasf Bony as results in this direction but nobody
I asked (incl. Brelot) seems to know. If there is anyone in Strasbourg
who knows please let me hear from him. By the way are you still there?


March 8, 1980

Dear Chung :
I didn't get any answer from Carmona, so I am forwarding the Note today. I am
not angry with him : it is difficult to reach him, since his university is Saint
Etienne, but he lives in Marseille, and besides that there has been a week of
holidays in between, just at the time I wrote him but anyhow, why delay the note ?
You are wrong about Carmona : he is no physicist, he's a browni.nmotion probabilist
like all of us, quite pleasant.
I don't like at all the idea of going to China, and I did my best to minimize
the time I'll spend there. As you may conjecture, this feeling doesn't arise from
antipathy towards China : it just means that the heavy atmosphere of respect sur-
rounding professors there is unbearable for me. But I couldn't say no to them twice.
I hoped very much Yor would come with me : it is quite impossible to respect him,
even for a Chinese- hets the kind of people every policeman in the world will find
disturbing, there are many amusing stories he told me about it. But.unfortunately
his wife's health doesn't allow him to go. You are entirely right about the lectures,
but anyhow I'll go very slowly, and anyhow there is at least one person which is
worth the trip, that is Yan.
I sent you some preprints last week.
With all my best regards,

'A lL

September 25, 1980
Dear Meyers

Your letter made me think how incredible it is that more than
8 years have passed since I flew from Paris to Shanghai on Air
France in june, 1972. I seem to remember vaguely that I sent you
a letter from Hangchow--- why I did is now totally forgotten.
It would be useful if you can tell people like Yan Jia-an that
if they want to go abroad to do research, the shortest path is to
work on something of interest to the people with whom they want to
works such as Yan has done judging from the three articles in the
last Semilaire. If such work were done in an area in which I am
presently interested, it would be a simple and natural thing for
me to invite him. No such case has occurred yet, but of course they
need a little more time.

Bon Voyage.




67084 STRASBOURG (France)
TEL. (88) 61.48.20

Dear Chung :
Yes, I am sorry you mentioned my name in your letter to Rao, and I sent
him apologies. You are rignt, but telling a sloppy writer he shouldn't be
sloppy any longer is telling an alcoholic he shouldn't drink. Simply, he
won't write any more. Personnally, I feel in sympathy with him since I am
a sloppy writer too.
About M.M. Rao, now : I never xsy said he was right ( and anyhow I don't
care about Orlicz spaces ). The Germans were terrorists, specially since they
demolished the same papers twice ( the first Landers-Rogge paper was more
than sufficient to warn against using Rao's results ). This is why I protest'
If this kind of papers got accepted in journals without control by the editorn
there might egen be pase g-of-- su.iicdpoa
-m' A ,_-- .... -L ... .b E g people prone to depression. In
the case of Rao I feel the guilt was Am the ZW, to have published it without
dropping the sentences meant to be offensive.
Of course Tor is charming On the reason why he is suspect to the
gendarmes, please ask them, not me ( and ask Tor to tell you his adventures
in Ulster ). I consider him, not only as a charming person, but as the best
am in the world in my own field ( though there are so many outstanding peo-
ple, particularly among the young generation ).

With best regards, and wishes,

1. I am not surprised at your report : I some letters saying I was wrong.
But they miss the point : in this << bad world >>, as you say, we shouldn't
let the outside violence inside mathematics.

J 5 May 5, 1982

Dear Chung :
My opinion about your book confirms the first impression : It is very
close to the book I'd have liked to write myself The story of books on
Markov processes is surprising : at the time the theory was quite down
to earth, books ( Dynkin, Blumenthal-Getoor ) were exceedingly general.
Now that the theory is almost dead from abstraction, the good concrete
books are coming out ( yours, Heyer's little book in German somebody
stole it from me, or may be I gave it to somebody-Murali's book also,
not to be forgotten, though I'll never write his family name GUo again,
nor dare to say it is a little QODUDIat places and finally, some day, we
may be happy to see Doob's book ). Of course, I am not meaning to put all
these books in the same package, but the tendency is obvious : young people
will get interested again in this theory.
Two concrete remarks : I strongly object to calling << projection thm>>
thm on
the measurability of debut From the point of view of the general theory,
this is a section theorem, and projection theorem has definitely another
meaning ( but the matter is not important ). Personnairy, I'd not have stres-
sed so much progressive sets : this is a seemingly simple notion, but full
of traps incidentally, as Walsh pointed out long ago, on p.37, line 5,
Tt should be [O,t[ not [O,t]. Optional is really better, and elementary
courses should contain Optional-Predictable, Ro*ar Progressive(nor Acces-
sible)which are for specialists. Why don't you say progressive ? It is true
that in French the difference between progressif and progressiste is neater.
The origin of the class D nomenclature isn't obscure at all I D means
Doob, of course. He considered the class of harmonic functions such that
for all open sets U, fljU with respect to the harmonic measure HU is uni-
fommly integrable family ( his Berkeley Symp. paper on Dirichlet problem ).
I simply translated this into martingale theory.
Do you remember I had a student whose work was to rewrite classical
brownian motion theory in modern language ? If he had been able to do it
5 years ago when I gave him the subject, it would have been quite valuable,
but he wasn't. Anyhow, the .work ( incomplete as it is ) may interest you,
and I'll send it to you next week after he finishes correcting misprints.
With best regards,

A v'u1CA.


UER DE MATHEMATIOUES ET INFORMATIOUE Strasbourg, le November 9, 1987
7, rue Ren6 Descartes
67084 STRASBOURG Cedex (France)
T61. 88 41 63 00

Dear Chung :
I should have answered you much sooner. Please excuse me. I didn't
see M6tivier to tell him that you didn't get you money yet from P. L6vy
( you said Yor, but he has nothing-to do with it ). If I see him I'll try
not to forget, hut anyhow I think you shouldn't hesitate to write him. He
won't feel offended. About Hu., I have been recently very satisfied by his
work, because finally we did something interesting together, which I'd
have been unable to do without him. He is returning to China in December.
The subject of this joint work is expanding in multiple stochastic integis
the solution of a stochastic differential equation ( this had been done
by two japanese, Isobe and Sato, under Ikeda ) and proving some non trivial
properties of the coefficients.
Martin came back from China, but he hopes to return there from time
to time. His spoken chinese is excellent ( chinese opinion, not mine )
and his written chinese has-been improving -since-he has to answer letters
in chinese now. He told me he will answer the letter he got from you at
Wuhan. At the present time he is also improving his spanish, however, since
he has decided to do mathematics in Madrid ( there is a very good group
of analysts there, all sons and grandsons of Calderon ).
Now let me come to the main subject of this letter : L. Schwartz is
very eager to get one of the big prizes of the french acad6mie for Neveu,
and sinee he and I are of the same opinion about Neveu, and Schwartz is
going to travel before the crucial meeting ( who is to take place on Dec.
14th ) he asked me to get support for him in this attempt. If you feel
you have something strong to say in favour of Neveu, please write it as
soon as possible either to me, or to Cartan or Choquet ( not to Schwartz
for the above reason ). 0a cl
With kindest regards,

erroJr ^- ~d~rua 2 SQUO O GsAC

'February 16. 1995-
Dear Meyer:
Your card was a surprise, but thanks. Zheng had telephoned me about
your retirement, but of course' I had known long since. In fact. Yor asked
me and I had meant to write a little esagy for you (and maybe also Neveu),
but we were both afraid the INCREDIBLE event could happen again,(fow which
you were partly responsible) and which r would not tolerate. Hence, rien.
What would I have said if I did write? Here were some of the anecdotes
which had passed through my mind: (1) -your sup martingale result which was
the first thing Doo'b told me about a French student in Urbana 'later he
gave a rnon-section(projection?) proof on the plane to Peking. (2) Your
lectures on the Jap [sic] duality later published and "sent" John Walsh
"to sleep,' These results obviously (to me) led' nowhere. (3) Your discovery
of the horrid mistake by icKean-overlooked by meA Watanabt and -not admitted
by McEean for six months/- I already published this in English but in China
---did'you see it? (4) Your mistaking M. M. for Murali Rao and later your
foolish defense of him against an innocent young German. lie did not say
anything illegal or unethical, and (I know) actually told the truth-nofet-
sho pologizen him. nhi, (5) One of your early letters (all
burned now) in which you complained that I had said so many unpleasant things
..... Quite frankly I never thought they were disagreeable! But I can un-
derstand one who regards Montherlant as "mauvais". .....
r. Ad you sees it was better that I did not write anything. However*
since I am in the middle of drafting a divertimentOand have been spending
an enormous am8;nl/rifling through ancient books (including your English
one), perhaps you can help by recalling- your own experience (in Urbana or
Seattle or before). How and where- did you first learn of these tribe FT+
for an optional T [< or < as you1 like]? It seems nowadays "obvious" but
why thhn Doob messed it up so horribly, see pp. 366ff of his 1953 book.
Was there really any difficulty there? If you can recall please tell mes
but don't tell me just what you know now --- that is not the "problem".
In my last letter to you I asked about Lyons and you replied some-
thing. I never heard about this again and it doesn't matter. My book will

come out soon. I despise the american way of saying "keep busy" after re-
tirements but it is true that 1'ennui is a problem. As you see I succeed-
ed in "killing" it at least for half an hour now. With kind wishes

Addendum on the reverse side

.'lUd 2 .1

(6) I remember that in an eWly issue of your S'eminaire, later deleted (?),
you discussed retournement du temps. That had something to do with my
writing that paper with Walsh, although I had known how to do it for a
Markov chain. Then came your rendition, Az'emas ..., Kuznetsov. I saw
the last-.iamed in Ithaca last summer* and asked him if he knew of any
VRAI emploi of R.d.T. to potential theory sucr as Hunt did in his
THIRD memoir (without r. d. t.) and later -others did with numerous sup-
plementary duality hypotheses. He did not reply but probably did not
know any (or care to know). This is a sad affair. So ny LAST questionn
to you, who started his career in potential theory, and wrote five volumes
on the probability part of it "pr. et pot.". IS THERE ANY? R.S.V.P.

( ) sans numero. OPEN PROBLEM. For Brownian movement in R-, X(nb6) is out-
side B.(Oil) for all large n, each 5. X(.) is continuous. Deduce from these
facts (and as little as possible more* maybe rien ne va pluA) that the same
.is true for X(t) for all large t.
This is a sample-of my divertimento.

March 27, 1995
Dear Meyer:

I THINK even a retired math'n has the obligation (to old "friends")
of REPLYING to simple, non-time-consuming, questions. So, to remind 'you

of my last letter and selecting ONLY the simplest ones:
(A) -id did P-A. Meyer learn of F T+? Read Doob 1953 to see how HARD

it was for him. I told him so and he freely admitted --- that was one VIRTUE

[Confucs] of Joe's that folks like 4IcKean, xxx, did not share!

(B) Did you know any application of reversing Markov process to the
caf'sical potential theory such as YOU studied under Cartan once? I asked
the same question of Kuznetsov last year. He did not answer. "C8est jolie

sis mais n'est point utile' --- my own words (hope my French is correct).

Similar word were spoken of your old lecyures on "Ito claculus" with jumps

--- here is a secret revealed. Your pal Kahane asked me that in Durham.
A new, iost curious question: who among the ancient greats observed
the follwoing connexion between Newton-Coulomb abd DeMoivre-Laplace-Gauss-

Einstein: ,.\" 'S

I hope this time you read my letter before writing a nice "reply".
!ith kind regards,


k 6/ .77

A6 y<

nkc /C. 1 T ?

June 2, 1996
Dear Meyer: D
Let G(xsy) be the Green function for a bounded domain in the plane#
then for fixed x (in Ds if one wisee), the function is upper semi-conti-
nuous in y. Of course there is a itore general statement about a more gene-
".AS. c-.
ral D and in (xsy). Persotne can prove this or disprove it. incl. Kesten.
Give it to a bright jeune homme (gille) if you know one. I don't think
Yor knows the answer. \W ce" T
You guessed quite wrong about that jeltne homme --- it is not you. You
also made an "inapt" comment about Hunt's (casual) remark on separability.
I was astonished that you would read this into the fine page I printed in
my book. Almost, though not quiep as astonished when I pound out your
behaviour in the L6vy affair [you lild to me about not knowing the merde]
As I told Yor to this day it is impossible for meto imagine a (nice)
lo w ')<(,b o u t ` L x>ly ,
person like you who knd how wrong th a a (about my examples)
and nonetheless on the feeble pretext of the "rules of the game" [what merdA
particularly for a Frenchman] my "invited", hidtorial reminiscences of our
< maitre would possibly be rejected because "it interests personnel [*ani 's
wotds translated roughly and after 9 years(J, You must beeout of your mind
Had it been you, and not Laurent who had (has) sense* I would have sued you
all in the world court which is trying to judge the Kara c types! By the
ways there is a HUGE [tremendously important for polytechnicians who as you
L -./ know go out into the real word as hadt-fonctiornaires] mistake in Laurent's
TEXTBOOK around p. 11. Twice I asked first Neveu then Yor to signal Lau-
rent and they both refused mumbling aqmething about Laurent's age (not so
old for me) and "poor health". Finally in a splurge of Onergy I did it
myself together with my @ag discovery of the first truly fundamental
-Ie of a "Schwartz distribution" --- see Note 1 of my gift to you* By this
time I hope you have leafed thru the body which is also full of "remarfues"
---some slightly mechantes, BUT NEVER SO MEAN as the behavior of some in
the Levy affair!. Not only he immediately thanked me but 4aid he would
correct the mistake in the next edition. It is very symp totics as in the
other affairs that NONE (French or else) had told him about it! It is now
clear that there is something fundamentally missing in YOUR French system
of science and education --- is it Napoleon who was responsible# ~2?
Tend your/ mn garden! I gave a vo\py to you but not Ja4ues, why? The
latter with pro ability 0.95 would not eyen ackwedge its receipt (on)
SA. Thank you for the quick ackowleg ment.
xxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx
7 In a sequel to Green* Brown, ... yox will appear in the Levy affair, be
warned. -* A+ If

UNIVERSITE UER de Mathematiques et d'Informatique
LOUIS PASTEUR 7 rue Rene Descartes, F 67084 Strasbourg Cedex
T~1. (33) 88 41 63 00 Fax (33) 88 61 90 69

le o10 juin 1996

Dear Chung :
About your question on potential theory, certainly Yor doesn't know the answer, but
certainly Choquet and Mokododzki know! My vague feeling is the following : GD(x, y)
must be jointly continuous except on the diagonal, and whenever x, y are close it is the
same as GB(X, y) where B is a ball containing them, up to a very regular correction.
Thus since the Green function of the ball is explicit everything is known! The answer
is likely to be in Frostman.
But all this is just a vague guess : I am now completely retired, and so happy I don't
care any more about all that (except that in the Levy affair I am deeply sorry that my
influence contributed to war instead of peace, which makes me liable to your accusation
of crimes de guerre. The only thing I can do is to apologize again, which costs little but
at least is sincere). I am now registered as a student a l'Institut des Langues Orientales,
in Bengali language, a full-time occupation (though I am trying to read also a few lines
of Chinese poetry every day). It is very refreshing to be a beginner again. In fact, I
feel I should never have been considered a serious mathematician (possibly not even a
serious person at all) if you happen to expose me in your next volume, I give you
herewith the written permission to use this statement (though exposing people without
their permission is more enjoyable).
I consider mathematics as a beautiful, but very tiresome business, possibly the
most tiresome after theoretical physics which explains why mathematicians are the
most outrecuidant of all people (think of Pascal, Descartes, and recently Grothendieck
and Thom having solved some mathematical problems they feel competent on any
I wanted to check the reference on the big mistake in Schwartz' book, but your
beautiful little book has been lent to Yan Jia-An, when he visited us recently.
With kindest regards, and wishes


UNIVERSITE UER de Math4matiques et cilniormatlque
LOUIS PASTEUR 7 rue Rend Descartes, F 67084 Strasbourg Cedex
T41. (33) 88 41 63 00 Fax (33) 88 61 90 69

le 10 juin 1996

Dear Chung :
About your question on potential theory, certainly Yor doesn't know the answer, but
certainly Choquet and Mokododzki know! My vague feeling is the following : GD(X, y)
must be jointly continuous except on the diagonal, and whenever x, y are close it is the
same as GB(X, y) where B is a ball containing them, up to a very regular correction.
Thus since the Green function of the ball is explicit everything is known! The answer
is likely to be in Frostman.
But all this is just a vague guess : I am now completely retired, and so happy I don't
care any more about all that (except that in the Levy affair I am deeply sorry that my
influence contributed to war instead of peace, which makes me liable to your accusation
of crimes de guerre. The only thing I can do is to apologize again, which costs little but
at least is sincere). I am now registered as a student a l'Institut des Langues Orientales,
in Bengali language, a full-time occupation (though I am trying to read also a few lines
of Chinese poetry every day). It is very refreshing to be a beginner again. In fact, I
feel I should never have been considered a serious mathematician (possibly not even a
serious person at all) if you happen to expose me in your next volume, I give you
herewith the written permission to use this statement (though exposing people without
their permission is more enjoyable).
I consider mathematics as a beautiful, but very tiresome business, possibly the
most tiresome after theoretical physics which explains why mathematicians are the
most outrecuidant of all people (think of Pascal, Descartes, and recently Grothendieck
and Thom having solved some mathematical problems they feel competent on any
I wanted to check the reference on the big mistake in Schwartz' book, but your
beautiful little book has been lent to Yan Jia-An, when he visited us recently.
With kindest regards, and wishes

August 9* 1996
Dear Meyer:
May I trouble you for some honest (brave) "critique" of subjects I do
not know and about which I do not know anybodyjelse who might give me an
honest opinion? Two of your ch&se associates ad told me that you tended
to take the side you were really opposed to, in order to show your unbias-
edness. The example cited then was your taking the side of the revulution-
nairea(soi,-d nts irca 19641) although your proper upbringing and general
comportement should place you squarely into the haute-borgeoisie. C'est
veai? That anal-masochism (a la Freud) could be one way to interpret your
behavior in the Levy'affair--- because you knew very well that that marde
was WRONG (since I taught you some Markov chains in 1967).
The question ist whether the *quantum probability" propagandized by
folks like AccaZdi ~4s anything but a brand of operator or functional theory
(he learned it from his maltre Gelfand), with of course some general mea-
sure theory (algegra really fpr les autres) thrown in as your pal Laurent
tried in his latter days (and you mildly criticized). Accardi was honest
enough to ask me to give a 2-hour lecture in his Univ. Roma II. in order to
"learn" (did he? I wonder! most math'ns don't learn from other camps) what
's all that in "Huntis theory. They (i. e. quantum) habe all sorts of har-
monic things (forms, maps, ad a) but whe n I did on the blackboard the an-
cient proof of the Dirichlet boundary result (see Green, Brown* and ... I
gave you, not in the Notes though) he confessed that he had never known it.
That is rather charming of him but it proved to me (biased) that their the-
ory had very little (tout petit peu) of y( "random time" in it (see also
loc. cit. Sec. 4). Later I heard a lecture in Cornell by your colleague
[to whom you were supposed to be politically opposed in 19671 Malliavin.
He had capacity all over the place but absolutely nothing random. ,After
the lecture I mentioned this to him, together with a corrected copy of his
lecture script full of the most remarkable English mistakes to the extent
that in one resul-t"he" translated a yes into a no --- he was gentile enough
to thank me and we had planned to discuss these matters on the next occasion
when he was (luckily for him) called away by his mothers illness. So to
expand my preceding question on quantum probability t little, what about the
Malliavin theory? Murali (NOT M. M; you put up in your dorm) told me it
was "just advanced calculus stuff"). Is that also just some messy measure-
p. d. e. without any randomness ingredient; I believe you have even writ-
ten something without tears on it (I neter tried to read it), so you ought
to know. But please this time tell it as it is and not try too hard to

be "objective" and "take the devil's side". You seemed to be quite brave
when you commented (perhaps unintentionally) on Laurent's cylindrical
measures osten4tily generalizing your ( ) probabilistic things. Of course
that is old man's efforts.
I would not have ELAen to consult you on the matter but for the in-
voluntary involvement forced on me --- but of course I can simply tell them
that "I donit know a thing". I am just afraid such a response would be
interpreted wrongly. By the way I did spend several hours trying to read
some of the long long papers on q. p. with hundreds of atomic-physical
terms and titles. That is how I reached MY CONCLUSION that it is NOT
"random probability"s viz. probability theory with random variables in them
that CANNOT be easily written down in other termsA but as I said above:
operator/functional stuff with of course some D-dimesinalf measure such
as Wiener's in it. By the way I anver can understand the true meaning of
that co-dim. (which you also used in your 'oher Jalks Isn't any med ire
constructed al la Kolmogorov always oo dim.?? If so. why must one say so?
After all* a topological space (Polish or any other junky) automatically
o dim a(ather dimensionless. Unless one treats specifically say Hilbert
space why should the dimension play a role? Brownian Motion in R3 is CI-
dimensional* isn't it? Correct me if you will. Merci beaucoup.

Bien entendu: to sa ha anZa 1l by le autres "quantum oat"
seems to me really not a at but a g is NOT ing the beauty of cats
and dogs. D'ailleurs, in the U. 8. there are about ten times op-func math
as (random) prob ---I do not know the situation in France. Cor. Accardi
and Co. must not be insulted if I say he is doing one thing rather than
another. Unfortunately many of our friends are small-minded and tend to
regard a classification as necessarily a rating! Please don't.
AjoutA le 13 agosto: Yor phoned me just now. He did not discourage me
from troubling you with my questions about q. p. So I am mailing it.

Nov. 25v 1996
Dear Meyer:
This will be more agreeable, I hope. But must begin with a
negative note: Feyel told me that Brelot -t al had apparently
considered similar (??) problem as mine about the u. s. .c. of
gD(x..) which aid Mokobozki certainly knew. He certainly did not,
and nobody does, ancore. Smettiti,-
I am going to talk about "Probability and Doob" in a few days*
After reviewing all his reprints I found that I had forgotten a good
part of what I myse had exposed of some of the stuff! As recent as
my Lecutess only 15 years ago. Here is one which I DARE NOT ask the
youngsters but perhaps you are old enough to answer. If h is (say
bdd) harmonic than h(X(t)) is a martingale where X is BM. The stupid
proof (given by Doob for submarts 1954) is to copute Pth(x) using
the normal distribution. So two ancient questions arose from my deep
(1) h=Pth but you seem to have written somei;wre as to the DIFPER-
RENCE between "invariantV and "harmonic". Is this an exceptional case
where they are identical. My bright idea is if we stop t to ma e it T
then of course the fact (under certain hyp, come toujours) that
h(X0), h(XT) forms a mart gives the representation of
h, as ball average* so harmonic. But if so any,other distribution that
is spherically (bally) symmtric will see through the same calculae- for
Qths.... Are there actually any crazy non-harmonic h to make h(X(t))
a mart? Repeat: in the calcul Doob made in 1954, any other distribu-
tion spgericealy symmetric wilj"work" (vulgarism---don't use!). But
I am sure there is no other except harmonic (at least if we assume conti-
nuity). Yes or No?
(2) Viceversa --- perhaps even more stupid: for harmonic h there
must be other X(t) to make h(X(t)) a mart* forget about side conditions
such as xMrintegrability .... Is there such a ch racterization:
(???) if ft certain harmonic fuentions (in Rd) x(t)) is
(say continuous) martingale, then X must be ....?
(3) This one is for any of your Chinese or Taiwanese students or
associates (no French will be interested): let Xk be i. i. d. and Sn
the usual sum. Assume Xl is NOT integrable* viz. both pos. and neg.
parts of it have infinite expectation. Assume.XIL is integre-valued
such that P(S n=a)> 0 for all large n depending on a [NaS conditions
of this are "well-known" but recently a famous probabilist admitted he

^~ ~ QL7 c ^^L

was unable to verify them after I told him! Then prove the elementary
old result: for any akintege any a in (0O1). we have the order

P(Sn=a)/ an goes to co.
Don't try it yourself but do test it on one of your students. I saw
that France was now rated mathematically-Lycee-grade more or less
with U. S. A. --- and England and Germanyl Ha-ha (that's why I only
suggested above Chinese or Taiwanese, assuming there is no Singaporean
under your direction nor South Korean --- they will all go to four
statistcis or computer colleagues* senon addiritura Bus. Sch. ,-*'-.
Among the French you are one of the best letter-writers, so
I hope you won't delay an answer too long. Wishing a Happy New Year.

You can be certain that I won't mention it but I am curious
to know whether (my) formulation of the big Doob Separability Theorem
r really appeared to you and Claude (: as a slight clarification of
Shis messy definition. Namely* to actually define a real-extended-valued
function of t,as "separable w. r. to S" and then to say that any roc-
ess can be modified so that allamxif sample functions are seprable (in
the new version). Not to claim any credit, this was first done in
my M. C. books 1960. J. Wolfowitz told me that until he saw that defi-
nition he never understood Doob's (in his 1953 book, open for inspec-
tion) The one you gave (first time I actually looked at it for my
talk on Doob) with Claude is a Bourbaki eion ( 'h). Do you know
that the possibility of demanding j "right separable sample functions"
was also observed by me, though only announced. By the ways did you
actually NEED it anywhere?
One regret (I don't believe your 5 tomes will have a re'Used
edition) is you did not give the "mirabile dictu" (see Green, ...)
definition of FT+ via FP a true beauty (and UN-expected) that Andre
Gide would have appreciated. In my talk if I have time I will/say,
as I did in Green, that poor old Joe never knew how to define his
FT in his continuous-time martingales. This is a marvellous example
of what an innovator can MISS. Please tell me your HONEST opinion
about these ancient quibbles --- by the time I hear it my talk ay-
long been gi~'e, 0Sevous pouvez etre sincere sans peur.

Jan. 12, 1997
Dear Meyer:
Thanks for your two letters. After I got the first I asked your
pupil Zheng weian to solve the little problem I asked you* but he
apparently was not able to do it, so I will repeat it here for you
or any of your associates to solve. Certainly it is not a hard pro-
blem and certainly it has a solution.
Let h be a harmonic fucntion in R d' d= 2 or 3, subject to some
growth condition so that on the one hand it is not a constant on the
othe hand if later we need to integrate it with somethings it should
have a finite value even afer we take absolute values. Now the questi-
tion: does or does not ex ist a semiggaoup Qts not that of the Brown-
ian motions such that h = Qt h ?? Make Qt iso--- if you wish to ap-
ply Gauss theorem. Doob did this for a subharmonic h in his much-
cited (how mu h read is in dou t) paper of 1954* but straightforward
cal/cul. If one (i. e. I) carry out the calcul one sees that one
needs only the iso... [I forget the proper word] and some kind of
integrability. very elementary. Hence t e question.
France is full of smart young kinds, even if you don't have any
under your wings* please find somebody in Strasbourg (how about Emery?)

"Y 6 k or Paris.
You say you don't do any mathematics now, then what do you do?

^6t ^ ^ H/e

le 5 mai 2000

Dear Chung :
Personnally I would try to prove this theorem as follows. Assume [0, 1[ is the disjoint
union of [ai, bi[, and set ai = ai A x, bi = bi A x. Then let E be the set of x such that
x = i(bx aif). your proof amounts to saying that 0 e E, that whenever ai e E then
also bi G E, and that E is closed from the left, and "therefore" (this is the place where
transfinite induction is used) 1 G E. I entirely agree with transfinite induction, since I
was educated that way before I knew Bourbaki, but it is clear that it is not necessary
here, because the set of all ai is countable well-ordered by itself and doesn't require
an external well-ordered set for induction. I don't know how Borel himself proved it.
Lebesgue quotes Borel for proving that if [0, 1] is covered by open intervals ]ai, bi [, then
Ei(bi ai) > 1, which follows from the finite covering case plus compactness. He refers
to Borel's Legons sur la theorie des functions. This clearly implies our result : add open
intervals containing the ai such that the sum of their lengths is < e, and then let e tend
to 0.
I no longer care much about errors in my books. They were not written for eternity.
They had their usefulness in their time, and this is already quite a lot.
I had forwarded your letter to Claude Dell. with a copy of mine, and I don't know
how many months it took to reach him. He may have thought he had nothing to add. e, c f
He may have thought so much time elapsed that it was purposeless to answer it any
more. After all, you and I are retired, but he isn't. Also, the letter was written to me,
not to him, and people don't feel obliged to answer other people's letters. Anyhow, I
don't see why not answering letters should be specifically French!
Yes, I heard at Besangon that Doob was ill, I got news from him through Catherine.
Since from your letter it seems that he is better and writes again, I wrote him yesterday.
With kindest regards,

A k.hM4A