Correspondence, 1971-1980


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Correspondence, 1971-1980
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Folder: Correspondence, 1971-1980


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Mathematics -- History -- 20th century

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Full Text

Lund, Jan 26, 1971

Professor Kai Lai Chung
Department of Mathematics
Stanford University

Dear Kai Lai Chung:

I still remember your short visit to Lund ten years ago with
very great pleasure, and I 6iten talk to my wife about your
visit to our summer cottage one Sunday evening you may also
remember it.

Now and then I have played with the' thought that you would return
here and stay for some longer time, and I think that my friend
Thomas Polfeldt asked you in general terms when he visited Stan-
ford two years ago and got an answer which he interpreted positively.

Now a direct question: would you care to come to Lund for acouple of
months next year, preferably during some part of the Spring term
(Jan May 1972)? Naturally, other people would like to see you, too,
for example Esseen in Uppsala and Harald Bergstrim in Gothenburg.
I guess that we can pay something in the interval Sw Cr 6500-7000
per month (divide by 5.2 and you get dollars) and for the jouney.
In return, we would like you to give some lectures and seminars each wee
on some of you favourite subjects. (In Sweden, professors generally
tech about four hours a week, but I mention this only to give you
an idea of the situation.)

I wish to add that Lars GArding was very enthusiastic when I men-
tioned my plan to him.

Perhaps I should also add that I have not at present any money for
this project lying in my hand, but I think it is best to have your
reaction first. Please tell me in your answer if you have any alter-
native time which you would prefer and also if the payment is satis-
factory or not in the latter case tell me very explicitly your

,-Sincerely yours

Gunnar Blom
Kvvlingevdgen 24a,
222 40 Lund

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CIENTfFICOS October 22, 1974

Professor Kai Lai Chung
Department of mathematics
Stanford University
Stanford, Calif. 94305

Dear Kai Lair

I had a glinpe of you in Vancouver but while waiting
for you to be free we drifted apart in the crowd.

It was good to hear from you and to know that you plan
to visit Mexico. Certainly we would like you to stop at our
University for a couple of days and give us a talk. Our Fa-
cultad de Ciencias offers an honorarium of 100.O0 U.S. Dlls.
for a Colloquium and 1200,00 dollars if the visitor stops
for a week with us.

There is a small hotel midway between the center of the
town and the University near Avenida Insurgentes, The hotel
has a restaurant but also a block of two from the hotel there
are quite a few good restaurants This hotel where many of
our guests here stayed has a nice garden.

Christmas, holidays at our University start around
December 20, you might like to be with us the week of December
16 20 and to be free after that to visit other sights in
mexico of which there are so many!

I shall be looking forward to your visit, meanwhile let
me know of your plans so that we may make hotel and other
arrangements if necessary.

With best wishes

Yours sincerely

Dr. Felix Recillas J.

P.S. Let me remind you that mail is very slow during the
months before Christmas. In case you have further questions
I would advise you to call up. 7y home phone number is
The Hotel L'Escargot is at the corner of the streets
Filadelfia and ;Oklahoma,

New York University

Courant Institute of Mathematical Sciences
251 Mercer Street
New York, N.Y. 10012
Telephone: (212) 460-7100

December 2, 1974

Professor K. L. Chung
Dept. of Math.
Stanford University
Stanford, California 94305

Dear Kai Lai:

Varadhan says he knows a quick proof of the Skorokhod result but
only using a big machine. He is interested to know what you have
discovered. Your systematic use of lost exit times is really new to
me, though as you know I thought about it a little. In particular, it
is my opinion that your method for the Newtonian potentials is the way
to do it. It will be nice to see what comes out from this point of
view, especially as regards local times. I guess we covered what you
wanted to know about Ito K. on the phone. It was nice to hear your


Henry McKean

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TELEPHONE 0223 61200 CB3 ODS

WRH/HG/MB 12 March 1975

Dear Professor Chung,

I understand that you may be able to come to Cambridge from
April to September 1976. If this is so, our Fellowship Electors
would very much like to offer you an Overseas Fellowship under
Title F for the period that you can spend in Cambridge.

The Fellowship will be supported by the Winston Churchill
Foundation of the United States, which is the American memorial
to Sir Winston Churchill, the Founder of the College. In
addition to financing the U.S. Overseas Fellowship programme,
the Foundation supports U.S. Churchill Scholars for advanced
study at Churchill College. It has also established an Archives
Centre here to house the papers of Sir Winston and his

All Fellows are full members of the Governing Body, which
meets four or five times a year, and they are entitled to the
appropriate rights and privileges as set out in the Statutes,
Ordinances and Regulations of the College. These include free
dinners and invitations to various guest nights. It is expected
that Overseas Fellows will take part in the activities of the
College and will have associations with appropriate University
departments. The only formal obligation is to be prepared to
give an Overseas Fellowship Lecture on your speciality (or any
other topic) to a general University and College audience.

We are able to provide you and your family with a furnished
flat within the College grounds, at a rent of about 25 per week.
Unfortunately the College is unable to offer a stipend, but is
very willing to support applications to other bodies. The
financial arrangements for your stay here will depend on the
extent to which you can obtain support from other sources. If
you are unable to obtain adequate financial assistance, I hope
you will let us know.

/ As

As regards travelling expenses, if you are unable to obtain
any help with these, you should apply to the Executive Director
of the Winston Churchill Foundation of the United States. His
name is Mr. Harold Epstein, and the address is: 17 Lyme Place,
Roosevelt, N.Y. 11575; telephone (516) 546-7364.

We should like to know whether or not you are able to
accept the offer of an Overseas Fellowship and, if so, for how
long you would be able to stay in Cambridge and which members of
your family would accompany you. It would be very helpful if
you could contact the Bursar, Mr. H. George, regarding the
arrangements for your accommodation.

We very much hope that you will be able to accept our offer
and to join our society.

Yours sincerely,

Professor Kai Lai Chung,
Department of Mathematics,
Stanford University,
California 94305.


April 1, 1975.


Dear Chung,
When your letter came, I didn't think there was any
chance of accommodating you, and now, having submitted it
to our department (or school, as people prefer to say here,
I don't know why) I have been instructed to tell you that
it is as I feared. At this time of the year, and the situa-
tion being what you know, all the "slots" for the coming
year have been allocated (and financial resources fully
committed), so that not one of the apartments available
for mathematicians is available any more.
Incidentally, while things are so tight, our school
is normally very reluctant to have visitors for only part
of the normal term (terms being from late September to
middle of December, roughly, and frop early January to
early April), since this ties up an apartment for only
parttime occupancy. Had you written even in February, and
been interested in staying for a full term, I am sure.the
""school" would have made every effort to accommodate you.
Presumably your answer to this would be that you could not
have written earlier, because your plans were too unsettled.
It's just too bad.
Are you at all likely to come East in 1975-75 ? Would
it be of interest to you to give a lecture here, and stay
for a few days, perhaps a week or so ? This, I think, could
easily be arranged. I should be very glad to see you again
some time.
With best regards
Very sincerely yours



April 7, 1975.


Dear Chung.,
I am very glad that there appears to be a fair chance
of having you at the Institute. For a number of reasons it
is hard for me to gauge probabilities, but there might well
be some possibility for the Spring term of 1976, and I would
hope that chances would be good for the Fall of 1976. Nothing
can be done, however, until we reassemble for the Eall term
(i.e. early October). In the meanwhile, you ought to discuss
the matter with Borel, who is currently at Berkeley; perhaps
he can gauge probabilities better than I could do myself,
even though neither of us is an expert in probability theory.
Anyway, you will do well to take his advice and follow it.
Here there is nothing that I coumd usefully do about it at
the moment.
Incidentally, I am due to retire in June 1976, but this
will.not involve any substantial change ifLmy mode of life,
except that I may have to shift from French wines to Cali-
fornian wines (we have almost given up drinking wine, anyway).
So, even if you come only in the Fall of 1976, I expect to
be here.


La Trobe University


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March 12, 1976 TELEPHONE: 996-3041

P. R. Halmos
Managing Editor, Springer-VeAclu
Undergraduate Texts in Mathematics
Mathematics Department
Indiana University
Swain Hall East
Bloomington, Indiana 47401

Dear Professor Halmos:

The book, "Elementary Probability Theory with Stochastic Processes,"
by Kai Lai Chung (Springer-Verlag Undergraduate Texts in Mathematics,
1974) is written in a rather charming style and contains many interesting
examples. It is all the more unfortunate that it is marred by rather
crude bias against women. One's sensitivities are aroused by the author
himself: on p. 115 a sample space of boys and girls is given as

{g ,gbbgbb}

followed by the parenthetical clause "[Observe that here we have ordered
the g's before the b's to allay possible criticism from Woman's (sic)
Liberation Movement,]" This contains the clear implication that the
Women's movement is concerned with nothing more substantive than the
order of precedence in sample spaces. (Why not a similar joke at the
expense of the Black liberation movement on p. 185, in the "racial mixture"

Perhaps the author needs to be informed that one of the concerns of Women's
Liberation is eliminating the sexist stereotypes which are used to keep
women in a socially and economically disadvantaged status. As an example
of how these stereotypes are perpetuated, we need look no farther than p. 144:

"A. little parable will clarify the arithmetic involved. Suppose
in two families both husbands and wives work. Husband of family
1 earns more than husband of family 2, wife of family 1 earns more
than wife of family 2. For a certain good cause...both husband
and wife of family 2 contribute half their monthly income, but in
family 1 the husband contributes only 5% of his income, letting the
wife contribute 95% of hers. Can you see why the poorer couple
gives more to the cause...?" End of parable.

No, I don't see why the poorer couple gives more to the cause. In fact,
this is false unless one makes the assumption that the husbands earn
substantially more than the wives--if, for instance, the husband-husband


P. R. Halmos March 12, 1976

and wife-wife differentials are equal, say x, then the husband-wife
differential in each family must be more than twice x. The author
evidently takes this situation so much for granted that he does not even
bother to make it an.explicit assumption!

One is therefore hardly surprised to find that in the text and exercises
all references to gamblers, customers, drivers, a "head of family" alias
breadearner (sic), Agent 009, election candidates, doctors, marksmen,
students taking tests, court witnesses, children bearing the family name,
and proofreaders are identified with the masculine pronoun. References
to females? A maiden picking the petals off a flower and murmuring "he
loves me, he loves me not"; a girl choosing a birthday present; housewives
who can win a sewing machine with coupons which come in boxes of detergents;
2 girlfriends of a man who visits them randomly; beauty contest statistics of
(36,29,38). End of list.

Exercise (a): What is the probability that the choice of sex references was
made randomly?

(b): What is the probability that the author is expressing
prejudices based on traditional sex stereotypes?

At a time when racist and sexist stereotypes are being removed from elementary
school texts, is it too much to expect the same vigilance from authors and
editors of advanced texts?

Sincerel yours,

Louise Hay
Professor of Mathematics

Copy to: K. L. Chung
Association for Women in Mathematics
Mathematical Association of America


Department of Mathematics

TEL. NO. 812
19 March 1976

Professor Louise Hay
Department of Mathematics
University of Illinois
at Chicago Circle
Box 4348
Chicago, Illinois 60680

Dear Professor Hay:

This is in answer to your letter of 12 March 1976.

Chung's "joke" about girls before boys is feeble and
not funny. Your reaction is violent and humorless.

To most people being "concerned with nothing more sub-
stantive than the order of precedence in sample spaces"
would seem to be on the same level, I should think, as being
concerned with the way the English language solves the problem
of the indefinite pronoun.

The purpose of Chung's parable is to make the reader
realize the possibility of statistically relevant hidden
factors. One example among several is based on the fact
that in our society husbands frequently earn more than
wives. Don't they?

Sincerely yours,

P.R. Halmos

cc: K. L. Chung


April 23, 1978

Dear David:

In answer to your letter concerning Hou, let me; begin with some
known facts. There was an article in a newspaper ]Guangming Daily]
last October praising his work. Names like yours and mine are includ-
ed, as well as Reuter's, as appraisers of his papers solving an open
problem. This showed that he had won some recognition, though probab-
ly not yet on the highest level. Otherwise it would have been in the
People's Daily! The fact his book had not yet appeared, after two or
more year's delay, is a bit odd. Also, the last time he wrote heom-
plained))that the American mathematical delegation did not mention any
contribution in probability. I wrote him in January suggesting among
other things a problem about Brownian motion)which I stated in my talks
in Peking (1975))as an example of the kind of things others are doing.
So far no answer, perhaps he is trying to solve it first.
By the way he is not a Dr. and is about 40 years old. In my best
judgment an award from you would make him very happy. It seems to
me also scientifically justified and polit ically innovative. Don't
worry anout the actual transfer of funds: money is not important in
China. He had asked me to send him some books and would be glad to
receive the prize in any form. I think he would also like to travel
abroad but that may or may not be easy. For instance, he must be aware
that another young man had been sent to Strasbourg to study probability.
and obviously he can greatly enhance his research by talking to people
in the field.

The only remaining consideration is human-political. I cannot
gauge how much intermural jealousy or snobbery there is among the
Chinese academics. Let us hope it is not as bad as in Russia. But
in a sense this is a proper time to do anything to help the cause
of scientists there because they are openly promoting basic research
and even learning from foreigners. There does not seem to be any ob-
\ C;in v vious rivalofor the prize. He is the only one who has written a book.
There was that newspaper article. All in all the chances are good
that whatever bureaucratic reaction there will be, it will take advan-
tage of the occasion. Perhaps Hou will be take, into the Academy, etc
To ask % e+i permission before making the award would be going too far
To consult the Academy as to how the public announcement should be made
may be the correct thing to do.

Sorry I am not acquainted with Mr. J. J. P. Deverill. Has he had
experience dealing with Chinese scientists? Hou's address is:

Changsha Institute of Railways, Changsha.

If there is anything else I could do, please feel free to write

C P y ur sincerely,

April 291 1978
Dear David:

You don't need any permission to cite anybody. Nor for any
criticism you wish to make. The world (of students) would be better
served if e. g. people (YOU inclusive) who cited any result from Ito/
McKean would point out the milliard of mistakes in it. On pp. 48-9
of its first edition, Durrett found a mistake in practically every
line. For remarks of a philosophical kind, as you also indulged in
once in a while, one runs more risk of changing one's own mind and
being too self-justifying. But to go to the other extreme and swaer
off one's own past does not strike me as healthy. Need I remind you
of your erstwhile agonization over Neveu and Ray? Whatever you say
now, it was incomprehensible to you then and Ray did make a mistake
about the discrete case (if I remember correctly). To answer your
question (43.iii), the answer is of course you will need more compacti-
fications as Walsh (and others) had already shown, for example if you
want to reverse. But again of course you don't'want to reverse. That
is your loss, not the theory's. cf ie noe~at
You seem to be the only British mathematician who does Ray processes
But still the oracular saying is: each in its proper place. To say e.g.
the early efforts on "approaching the boundary" ["Listening to the cucko
---by Wang Weill have been supermded by some compactidation or another
is just plain foolish. You should read the preface of my little red
book again, so long as you cite my wise sayings, also the introductory
remarks to my talk at the Nice Congress. If I had been a bit more per-
sonal, I could have added the following anecdote at Dynkin's expense.
He "criticized" my sample fundtion discussion by insisting on having
those miserable "characteristics". But in his own paper (of course
I never read it) he made a big mistake which was only corrected by my
student Pittenger. ]Both in the Russian J.] the correction is obvious
from the uncompactified view and after it is done, trivial identifica-
tion was made by P. between those characteristics and simple boundary
properties. I always had a sneaking suspicion that you had got into
the same kind of trouble with your matrix transformations, which you
never unscraadied and which you could have avoided by a bit more under-
standing of the truly simple structure of the boundary. The mroal?
Refrain from bulldozing your vegetable garden.
I look forward to seeing your tomes. But here again a little advi
if it is not too late. Your uxxexm articik "Path decomposition..."
showed a lot of "scritas reservatus". Freud said wisely that one did
not mind one's own stach, of which the Corollary: Expect not the reader
to do your exercises.
I am sorry that due to the return of our ill colleague there is
no room for your student here next year. If you can get a leave, wou
you like to visit and teach from your book?




December 11, 1978
Dear Jacques:
A telegram has just arrived from the Academy in Peking saying
that you* Doob, and I "are welcome to visit Peking". This means
that we can go as planned, but further details and a formal letter
of invitation remain to be negociA. I will make up a formal list
(perhaps adding a couple of names) and send it to them as soon as
I hear from you. If you are going with Monique please so indicate.
I phoned Doob and we thought May 15 may be a good day for us to get
there. If you have other suggestions please specify.
I will tell them that we will make ourselves available for all
kinds of lectures and consultations for two s three weeks, and re-
quest that we do some tourism for a week to ten days afterward.(some
of which may be combined with visits to provincial universities, come
on fait sur le midi!)
In the meantime, Merry Christmas and Happy New Year.

Yours ever,




Jan. 4, 1979
Dear Jacques:

I am glad to get your wire yesterday. It is formidable that
Monique is coming! A few days ago I got a letter from Meyer saying
that he had heard about our trip. I wonder who told him that. Per-
haps you have got a direct cable from Wang Shou-jen as Doob and I did.
Anyway the thing is now public. According to Hou (whose winning of
the Davidson prize was an event in China, and who was instrumental in
Fpgxm~xxg transmitting my proposal), some more pepqie may be added.
To make it truly international I am sounding an Englishmane.
Time is shorter than we think, so let maumake the following pro-
posal. Each of us should give a series of lectures on one topic.
Doob will probably talk about his book on Classical potential theory
and Brownian motion. I want to talk about some new results in more
general potential theory (of Hunt processes) which is joint work with
K. M. Rao (the one in Aarhus). So it would be nipe if you could talk
on a rather different topic such as Gaussian processes, unless you have
another preference. Assuming each session lasts under two hours Jxm
with translation (you can spaak French since Yen Chia-an studied in
Strasbourg and will be there), how many sessions do you need? Besides
these special topics we should give a few general educational talks
xfxpyax such as easy surveys of various areas, There will also be
round table type of discussions. If you can prepare for 6 to 8 hours
of talks without translation time and at a lower speed thar. your usual
speed, this should be ample.
We should try to arrive in Peking on the same day or very nearly
so. Air France has a direct flight to ShanghA (which -I-toek-in-1972)
aefd- ehas to Peking, but it may not fly everyday. If by May there
is a Pan Am flight from San Francisco to Peking I will take it but
otherwise I may well take the Swissair flight from New York via Zurich
and Geneva (and a couple of other cities,a to Peking. Apparently Doob
will take this one. When we decide on these flights please inddrm
each other. It may be necessary to wire Wang about the exact times of
arrival to be met and taken care of. tou and: Monique can exit wherexve:
you like but I adtive that we arrive directly in Peking to get the best
reception. Namely, Peking should be our port of entry.

Please let me know your plans ,intentions and suggestions. I have
to send Wang a detailed program wheri we know what we want to do.
Happy New Year.

As ever,



R&mistrasse 101
Telefon 01 326211
ETH-Zentrum March 30,1979
CH-8092 Zurich

Dear Professor Chung:

we are looking forward very much to your visit from May 7 to
May 13! I will make the reservations, and you will certainly be
met at the airport. Please wait if for some reason you don't see me

We have scheduled two talks, one on Wednesday 5:15 p.m., one
on Thursday afternoon (probably the same time). The first will be
in the general ZUrich Mathematics Colloquium (ETH + University).
So there will be the usual Colloquium problem that most people in
the audience won't be probabili'sts, and it would be excellent if
this talk would be rather expository, with emphasis on the general
ideas. The second talk will be in our probability seminar, so there
is no problem. Could you send us the titles, if possible 2 weeks ahead?
As to our excursion, we will have proposals ready when you arrive!
In the meantime best wishes for the preparation of the China trip,

Sincerely yours


Institute of Mathematics
Academia Sinica
People's Republic of China

April 5th, 1979

Prof. K. L. Chung
Department of Mathematics
Stanford University
Stanford, Ca 94305

Dear Professor Chung:

Prof. Wang Shou-ren told me of your willing-
ness to come for a visit. On behalf of the Institute
of Mathematics, I would like to extend our formal
invitation to you and your wife for an academic
visit to China for four to six weeks.

For the itinerary, the first three weeks will
be spent in Peking in seminars and discussions with
audience from Peking as well as from other parts of
the country. The rest of the time will be allotted
to visiting other cities.

I am looking forward to meeting you in Peking.

Sincerely yours,

Tien Fang-tseng
Deputy Director
Institute of Mathematics



June 28, 1979
Professor Wang Shou-ren
Academia Sinica
Beijing, China
Dear Professor Wang:
At your request I gave a brief oral examination to 4l~-4L, j
during my stay in Beijing in May. They were asked some questions
in analysis and probability, and their ability to understand and
speak,.'English was observed. As a result I am satisfied that they
are both qualified to become first year graduate students in U. S.
universities. The-r knowledge of the subjects tested is comparable
to students admitted to Stanford graduate school and they should
have no difficulty attending courses given,\this department. '
English is quite adequate. f- is more tidd and should have
more practice in using English.


Professor of Mathematics


September l, 1979
Dear Lars,

It is good to hear from you and that Eva has finally achieve
her ambition---immediately I recall the scene at the Cathedral..
Bergman's .. Stawberries.

Graph theory has grown very b6g under the pretext of computin,
sceince and lots of federal money. The people in these things art
very cliquish and one cannot expect any objective appraisal of the.
kind of work. I can't think of anyone who can evaluate the MS you
sent me and certainly cannot read it myself. I will hold it for a
few weeks to see if by chance someone drops in my office who can giV
me an opinion of it. Your best hcest way of refusing its if that is
what you wish to do, is to tell the truth that Acta math. does not
want to expand into these newfangled areas. ]You know/Acta math. witl
lower case m.?] why

We (I and Murali Rao in Aarhus) have solved the problem I asked
you sometime ago, very smoothly with probability methods. We can also
do the Dirichett problem etc. for the so-called Feyman-Kac semigroup
whose generator is the Schrodinger equation, without any extraneous
conditions. As in the classical DP things are easy under heavy condi-
tions, but we don't know even that has been done before. If you are
really interested and still ha the energy to learn a new trick or two
we can send the finished MS to you. One has to know Brownian motion
in R to understand the stuff, and not just Wiener space! I doubt you
have anybody in Lund for that. I jaut came back from 5 we ks in China
(many cities as far as Kunming where I lived during the war), and one
week in Switzerland. That week rounded up the trip very well for me.


September 3s 1979

Dear Marks

Very recently K. M. Rao and I obtained the result below which
you will recognize as your brainchild. Let G be a bounded domain
in any Euclidean space q bounded measurable in G, f continuous on
6G and nonnegative, T = first exit time from G. Put

u(x) = Ex(expV1T q(X(t))dt.f(X(T))}
Theorem. If u is not identically infinite in Gs then u is continues
and bounded in G. Furthermore if x in G approach a regular boundary
point z then u(x) conveys to f(z). CTI past leaves o ftrce !1
N. B. q may take both positive and negative values, no condition
whatever on the smoothness of 6G otherwise as stated (z regular in the
usual sense), and the boundedness of u does not depend on any regulari-
ty of the boundary.
This was a pleasant surprise to us, but we wonder if you know any-
thing of the kind in the literature. I looked at Stroock and Cie)elskis
They studied your potential theory in some depth, but they did not study
the potential with your functional' But there may be other work of
which I am unaware. Please drop me a line very soon as I intend to
write this up togethersome cute applications to positive solutions of
the Sohrodinger equation. Incidentially of course you know the u above
is a weak solution in any cases and a pointwise one provided q is also
Holders butas you used to say, "that is another story".
The proof is not too hard but rather tricky, not the kind a mere
analyst can get. I saw there was a recent revival of your functional
by mathematical physicists and analysts, but what we have examined
(papers by Cvrmona) does not seem to contain much of value from the
probabilistic point of view. Viva PROBABILITY, a bas operator! Hope
this catches you home and with best regards.



September 10, 1979

Dear Jerzy,
It was nice to hear from you. The task you asked me to do is
a pleasant one but not very easy. The inclosure is the fifth version
I wrote and I hope it will do. Please feel free to revise it together
with David Blackwell. Please send me a final copy, in case Jimmy Carte
speaks some of my words.
I have marked 20 items on the list you sent me, including a new
item which is the book Doob is writing. Some of the items are referred
to bj numbers in the narrative, but if you think this is too technical
these may be omitted. Db you still want a seconding letter from me?
If so please let me know. I have used up the proper words in the nar-
rative but I ca.n say something as follows. "Doob is clearly the fore-
most probabilist in the world after the death of Paul Ltvy." A word of
explanation may be needed for any mathematician who is on the president's
committee. Kolmogorov is an earlier master of the thirties and a.broader
scientist, but so far as probability is concerned Doob's work is more
substantial and influential for the present and the future. Such a com-
parison is of course inane, but some "decision-makers" demand ratings.
Finally, now you should plan that trip to Ciinal


P. S. Whoever compiled that list made a mistake in item No. 66.
which I corrected. There may be others.




September 2., 1979
Dear Hahs,

I ,ust got a letter from Masao and realized that my last letter
to hinm about the Oberwvlfash meeting took almost two months to reach
him. Also he is not returning to Zurich for sometime yet. I have
not relpied to Barner because I thought -P'or you would communicate
with him about the possibility of change of dates. The dates set by
huim is March 15-21 1981 which -:mn&ud by difficult for me since it is
the period when we have classes, ever examinations. It is perhaps too
late/to get any other date in 1981. There fore I thi~k Maso's sugges-
tion of September 1982 would be much better. Septe:,Mber is a month in
which? I am quite free from obligations and it is also a pleasant time
to travel in Europe. I wonder if you can telephone Barner to say that
we are still discussing the dates and maybe ask him to reserve a week
for us in Septmber 198? anyway. Please send him my regards and excuses
for not having replied so far. Incidentally I think I told you that
there is a probability of having a conference in China in 1981.

'Did you or N. retrieve that Chinese book from the hotel? If so please
send it by surface mail to me, many thanks. Here is a very interesting
problem which I did recently. It looks easy but not so. You wan try
it yourself and give it to your good students to try.

Let,,q be a bounded measurable function, G an open connected bounded
set in R (for any d) T the first exit time from G. X the Brownian motion.
Theorem. If standard
EX{ exp q(X(t))d J is not .identically co
in G, then it is bounded in G

This requires no condition on the boundary of G. Furthermore if
the boundary is regular in the usual sense, then a's x tends to the
boundary theexpectation above tends to one. There 'are of course extebc
sions of this in the classical manner, for example the first boundary
value problem with that exponential functional. Althoug.,the latter
is wellknown as Kac-Peynman functional, it is strange that nobody has
studied the associated Dirichlet problem etc. [so far asl can find
out]. The results should yield a probability theory for solutions
of the Schrodinger equation, which I am beginning Ito-do with Durrett
here. From the axiomatic potential theory point of'view, it means that
the Brelot-Bauer stuff may be satisfied by the equation (and I suppose
for othe partial d. e. reducible to the above). I believe such work
has been done by French :-nalysts but I don't thibk they have done it
the way described above. If you happen to know anything about it pleas
transmit. .^ .,^-6 .,




Dear Dan,
I am writing to consult you on p with which you are well
acquainted,. ILet X be n-dim Brownian motion, G a bounded open connected
set, T th.- first exit time from G, q bounded measurable takd~g-both +
and values, and I
u(x) = E( exp fT q(X(t))dt )
The6rem. (1) Either u is identically infinite in G, or it is bounded
in G, hence continuous. No condition on 5G.
!2) In the second case above, if x tends to a regular boundary
poikt then u(x) tends to 1.
This is thepristine .form for cognoscenti like you. Dirichlet, Poissoh
problems follow, and Schrodinger is not far behind.
These re ults came to me as a big surprise. One would not expect
u to be I)ounded up to (but nt-nd i oi) the boundary, and the Kac
functional has no 'effect on the boundary limit. sTeither :ark nor Doob
knows anything about its and I can't find anything of the sort in your
and Ciesdlski'sebut I don't know the literature. If you know of any
lease ro-p) me .f few lines soon as I am about to sit down to write it
up .(with ;R o who simplified a proof). We are alko pursuing many other
po.sibili.fts. A potential theory with the functional seems possible.
Incideiftally I don't have Part 2 of your paper on Kac potential
theory/. Ca a you snld it? If you are interested in the stuff we may
send the "11 to you, f we don't have to pay page charges. Iast.but
maybe least congratulations on the chairmanship!



November 16* 1979
Dr. William H. Pell
Program Director
National Science Foundation
Washington D. C. 20550

,Dear Dr. Pell:

In response to your request for evaluation of the special
project on the inclosed sheets I choose to write this private
letter for your consideration. This way you will not be forced
by federal regulation to show this verbatim to the proposer.
I had an extremely annoying experience recently when somebody who
identified my typescript (which was sent him by your office) and
phoned up to cuss me out. Since I value my own privacy I do not
wish this kind of thing to recur and so must request you to hold
this against any such intrusion.

Princeton is an excellent place for mathematics, at least in
.some areas. It has produced many good people. There is no ques-
tion that it can continue to do so. Therefore it seems to me that
the pertinent question to raise about the present proposal is: does
it need such a special project to carry on? I talked with some people
who seem to know the story. Apparently this proposal is a kind of sub-
stitute for an Institute. Berkely is making such a proposal. Prince-
ton cannot get it perhaps because there is already an Institute in the
same town. So Princeton wants a consolation prize because some of the
more aggressive types (Kohn is named by my informant) want more money
and power. What will they do? They will invite a few "big shots",
make NSF foot half of the bill, and then appoint a larger number of
younger people around these, again let NSF foot part of the bill. But
these things can be done under the present setting of NSF, through re-
search contracts and grants, and through special conferences. What
Princeton wants is to hkve more of these for a longer period, and pro-
bably with less control wnce the money is granted. In short, they want
a bigger out of the pie. Is it worth it? For certain peoples and for
certain areas of research, this bigger support may produce more results.
Some of it will of course be wasted in unnecessary expenses such as tra-
vel. Some "buddyism" cannot be avoided as it is well known that there
is a strong mutual admiration society among the VIP---you invite me and
I invite you* etc.
With inflation that high, I do not know the latest picture for NSF
funding ,fr mathematics. This will have to be an important considera-
tion fo/q/any (uch project siphoning away a lot of money. For another
reviewers Isuggest J. L. Doob at Illinois. He is relatively conserva-
tive bn these matters. Incidentally, as far as my own area of interest
is concerned (also Doob's), it has been dead for many years at Prince-
tion. -'So we can claim some personal disinterest in the matter.


5-'6" 7 W'.. ,Foland
Mll. 43-10-76
Wroclaw, November 28, 1979.

Professor K.L. Chung
Department of Mathematics
Stanford University
Stanford, California 94305

Dear Professor Chung,

Thank you very much for submitting your paper "Equilibrum
and energy" with M. Rao for publishing in the Journal of
Probability and Mathematical Statistics. I have the pleasure
to inform you that this paper has been accepted for publica-
tion in the second issue of the first volume. According to
our plans this issue should appear in print already in 1980.

I hope you have received my cable regarding the abstract of
your paper. We would appreciate to receive the abstract as soon
as possible.

Cordially yours,

imierz Urbanik
Editor-in-Chief of Probability
and Mathematical Statistics

SUniversity 1230 YORK AVENUE NEW YORK, NEW YORK 10021
O 1901 S

December 5, 1979

Professor Kai Lai Chung
Department of Mathematics
Stanford University
Stanford, California 94305

Dear Kai Lai,

Many thanks for your nice letter of November 9th and my
apologies for being so late in answering it. I have been out of town
a number of times and then I was swamped with all sorts of trivia.
I hope that you will understand and forgive me.

Rumors about an anniversary volume have reached me, but
knowing how slowly these things move I do not anticipate anything
happening until I reach the age of 70 if I live that long. I am,
of course, delighted that you will contribute a paper to the volume
especially on a topic which is close to my heart.

Now to your question: I do not know of any connection
between the equilibrium distribution on the surface of a conductor
and the stationary distribution of a suitable Markov Process. In
fact I am somewhat surprised that such a connection exists and
would be very much interested if you could supply me with some

I am afraid that I cannot take the credit you would
like to assign to me on the discovery, or what has become known
as the Feyman-Kac Formula. I am enclosing a page from my auto-
biographical note which may be of interest to you in this

I expect to be for a few days at the meeting of the
AAAS in San Francisco, and then I am scheduled to give one or two
talks at Berkeley. All this will take place between January 4th
and 10th, and my schedule will be extremely tight. However if
I can find a little time I may drive up (or is it down?) to
Stanford to see you.

With best regards to you and your family, I am,



1 + 3fJP(y)dy
K' 3Si P(j)

Julian's answer was the same except that in the formula for Ki, 1 was
missing! By that time I was both curious as to the origin of the discrepancy
and expert enough to spot it in the vastness of Julian's manuscript. It was
simply due to the unfortunate British custom of writing the indefinite
integral as

Is .

In copying one of Watson's formulas Julian set the lower limit to be zero,
and it wasn't! Thoughtlessly, I failed to acknowledge Julian's help with the
problem in [37]; but perhaps it is not too late to rectify this omission.
Julian's unmatched prowess as a classical analyst is, of course, too well
known to require further corroboration; but the feat of solving my strange
problem in a few hours (which also included a lengthy writeup) must surely
command admiration. -
All this brings me to a question raised in the Commentary (and which I
must confess haunted me from time to time) as to whether I could have dis-
covered the Feynman-Kac formula already in 1945 had I known that the
eigenfunctions and eigenvalues of the Schrodinger equations with attractive
potentials zlxl are related to Bessel functions of order 1/3. The answer is
almost surely no. I was just not thinking along lines that might have led me
to suspect a connection with the Schr6dinger equation. I might have been
forced into discovering this connection if I hadn't had a way out through the
availability of an explicitly solvable integral equation and the invariance
principle. If I had thought more and calculated less, if I weren't such a
devotce of "catch as catch can" method of solving problems, if, if, if, .. .
The fact is that if it weren't for Feynman, it might have taken years before
the connection would have been discovered; and I might have been blamed,
with considerable justification, for having obstructed progress by lengthy
and opaque calculations with Bessel functions. As it was, as soon as I heard
Irynman describe his path integral approach to quantum mechanics during a
I, ture at Cornell everything became clear at once; and there remained only
.-r..tcrs of rigor. _




Dean Dillon E. Majother
Graduate College
University of Illinois
Urbana, IL 61801

Dear Dean Matother:

I am writing to you at the request of my colleague, Professor Paul T. Bateman,
in support of the nomination of Professor Joseph L. Doob for the Wolf prize. It
gives me great pleasure to write you about Professor Doob.
Professor Doob is a master builder in science with a long vision. His work
is distinguished by great originality and inexorable force of logic. Early in
his career, and almost single-handedly, he laid the foundation of stochastic
processes with continuous parameter, a groundbreaking achievement heralding the
enormous advances in the field made in the following decades. Later in his book,
"Stochastic Processes", he introduced the simpler notion of "separable modification"
which remains to this day the only viable theory. He began his penetrating analysis
of martingales (and submartingales) soon after P. Levy and Ville defined the object,
but it was Doob who established nearly all the basic theorems and foresaw the great
variety of applications. An example par excellence is his proof of the law of
large numbers, the centerpiece of classical probability, by a backward martingale.
The methods of martingale theory have now become an indispensable part of the basic
probabilistic thinking in all its scope. It is capped by Doob's Stopping Theorem
which has its origin in the impossibility of a winning system in a fair game. This
is an instance how he recreated a new mathematical theory out of an old theme, a
folklore in the history of science. What is unexpected is that the same thread
of thought leads to deep and rigorous analysis of sample functions (trajectories),
on which the modern approach to stochastic processes depends and of which the chief
tools were fashioned by Doob. A vital area of application is the theory of Brownian
motion and the associated potential theory--the Newtonian theory of gravitation and
the Gauss-Dirichlet theory of electromagnetism. Doob wrote a series of fundamental
papers which laid the groundwork of a probabilistic potential theory encompassing
the classical case but going far beyond it. Here again he reshaped an old theory
into a brand new one. This was further developed by G.A. Hunt and P.A. Meyer
into one of the most flourishing chapters of contemporary mathematics. It is said
that probability theory ended its (splendid) isolation largely owing to this
development, in which Doob played a key role. His devotion to a more special
topic, a stochastic version of the Fatou boundary limit theorem familiar in
complex analysis, led to many contacts with various abstract theories of boundaries
which continue to be a major subject of research among analysts. In his forth-
coming book on Brownian motion and potential theory, all his life-long pursuits,
from measure theory through martingales and Brownian motion to.potential and
boundary theories, will come together to bear witness to his magnificent and
long-lasting achievements in this vast area of scientific endeavor.
As you know, Professor Doob has just won the National Medal of Science.
In my opinion he deserves the Wolf prize more than any other mathematician.

Sincerely yours,

Kai Lai Chung
KLC:ca Professor of Mathematics

Feb. 20, 1980

Dear Congressman Anderson,

I like your idea that we should revitalize our railways and
shunt federal money into them rather than highways and autos.
This should give a lot of jobs all around the country, make people
see each other more, less egoistical, less pollution, less accidents.
etc. etc. I had thought of writing to the New York Times long ago
but am pleased to have a presidential candidate voice this wonderful
idea. Why have people not thought of it? THE AUTO LOBBY2 Do try
to speak more about this completely realistic scheme. Many European
countries did it, why.can't we? We have such a big and scenic country,
yet we can't get to most parts because of lack of public transporta-

Since Eugene McCarthy I have not given much money to candidates.
I inclose an initial support with more to come if you speak out
more on these "impossible" solutions to our problems. Beat of luck.
[I am a registered Democrat but will vote for you if given the change.


K. L. Chung
903 Lathrop Dr.
Stanford., California 94305


Narch 13, 1980

12, RUE CUJAS, 75005 PARIS
TEL. 325.24.13 POSTE 361

Dear Chung:

yIr letter of November 29, 1979 reached me when I came back
from East Asia (Manila etc.) a short while ago. In the mean-time
I was informed by Barner that Jacod and I are to use the period
in March '81 for the Stochastic Analysis conference.
The Index volume for ZfW had already been decided upon and
planned a long time ago, see the foreword to vol. 50.
Bernoulli Society, advantages: in addition to the newsletter
"International Statistical Information" you will get, from 1981 on,
the reorganized Bulletin of the ISI, to be run by Barndorff-Nielsen
which will no doubt be very interesting, nothing to do with the old
More important: the satisfaction to help! During the last two
years, we have started the Latin-American and the East-Asian regional
groups of the Bernoulli-Society who will have their first meetings
this year#. They will certainly helpful for the development of
probability and statistics in these regions in many ways, by
informations, contacts, regional meetings, colloquia etc. E.g. while
in Manila I had a long discussion with the Chinese delegation to
the ISI-Session. They are in principal very eager to participate.
However, to do all this, we need a strong "core" of the Bernoulli-
Society in its traditional strongholds, i.e. North America, Europe
and Australia-New Zealand. So, how about it?
Hou Zhen-ting recommended Rossi's paper for ZfW; what is your
opinion? He also sent a manuscript, counterexamples to theorems of
Soviet people, naturally.
We could not find a recipe for Campucha noodles; do you have
Best regards, also from Angela



Professor Wu Wen-tsen
Institute of Mathematics (Computing)
Academica Sinica
Beijing, CHINA

Dear Professor Wu,

It has.come to our attention that some articles published in the recent
issues of Acta Math. Sinica and Kexue Tongbao are quite obviously trivial. Not
only are the results well known and can be found in textbooks, but they are
very simple consequences of easy theorems in books which are available in China.
Since these results are stated in the English abstracts, it is clear that the
authors lack general basic knowledge of the fields as well as a modicum of self-
criticism. But the responsibility must be shared with the editors and referees.
Continued publication of such articles in the major Chinese scientific journals
would create unfavorable and false impressions of Chinese research efforts. We
believe that a strictly enforced refereeing system by active mathematicians is
necessary to avoid such mistakes in the future. This should form an essential
part of the research and education advancement in China. We suggest also the
possibility of starting a new journal at a level somewhat lower than that of
American Mathematical Monthly to be the proper vehicle of publishing articles
of pedagogic interest to students and faculty. Even there an adequate referee
system will be necessary.
We are addressing this letter to you as the only mathematician on-the
editorial board of the Acta. We understand that as editor you cannot control
all the articles published there, and occasional lapses occur in all journals.
We wish to stress the fact that a small sample of articles does not reflect
on the general status of your journals, nor do we think that individuals be sou
penalized by their past errors. We are confident that with the progress of
time many of these authors will produce good research results. Our sole
purpose in writing this letter is to alert you to the existing conditions and
to recommend positive remedies. This letter is sent to you alone for'your
consideration since we all know you personally and have high regards for your
work. As colleagues concerned with the development of mathematics in China,
we expect a prompt and informative reply 'from you.
With best wishes,

Sincerely yours,

',-,c,7 c^A;^ 2 *^,

Prof. Dr. Hans G. Kellerer (VERMITTLUNG23941)

Professor K.L. Chung
Dept. of Statistics
Stanford University

Stanford, Cal. 94305

Dear Professor Chung,

Thank you very much for your letter of April 13 and the enclosed paper,
which I first transmitted to G. Winkler. Concerning your "silly question"
you may have forgotten some additional condition; otherwise here is an

Example Let C C a Borel set which is not of type F,; then
(0) O, if B c C .-
A e4 otherwise
S0, if B C U F with closed sets F C C
V(B) := M n n
L otherwise
defines measures on the Borel 4--algebra with
(1) /a(B) = v(B) for all closed or open sets B,
(2) ,(C) = 0 and v(C) = M.

Of course,-I would be glad, if you could arrange a visit to Munich during
your coming trip to Europe. I am away until the middle of June, but our
summer term closes only at the end of July. Please, let me know the
possible dates early enough to make the necessary arrangements.

Sincerely yours,

June 2, 1980
Dear Hou,

A few days ago I got the letter from Rao including your and Liu's
paragraphs. I am glad to know that you had a good visit with him. He
told me that Liu's translation was understood by all the audience and
they liked your talk. i hope to hear from you about your visit to Eng-
land. Can you get Kendall an invitation to visit China? He wants very
much to do so.

Thank you for sending me the old Chinese book and also the MS of
your new book. I have not read it carefully but the level seems right.
Since you want to give the book to Wiley I must tell you that you need
permission from other publishers such as Springer-Verlag for the results
taken from books published by them. It is all right to use so0ie mate-
rial in other books but it seems the cust omn China to use large por-
tions without acknowledgment. This is"ego. ble if the book is for
Chin-se readers only but if it is to be published in English it may be
violation of the international copy-rights convention. Even if China
at present does not join the convention iiley must follow the rules.
So you should get advice from experienced people before having your MS
published in English. I have told Wang T e-kwen the same problem because
his book on birth and death processes al4a contains many pages of mate--
rial essentially the same as in other published books.

I hope you are now working on so;ne interesting problems. Has there
been any evaluation of the work on reversible Markov chains done in
China? TUntil some papers are published in the foreign journals it will
not be easy to get attention. It seems that the refereeing system in
China is not adequate. Otherwise the trivial papers by your friend Liu
Wen would not have been accepted by Acta and Tongbao. Such articles in
the major journals in China would cause poor impression but more serious
is the question why he was not informed that those results were trivial.'
You should discuss this matter with Yang to whom I wrote a few
days ago. People likh you and Wang T.K. should be responsible for such
occurrences&' It is a pity that Liu wastes his talents on these problems.
Did you not know that these were immediate consequences of elementary
probability? It is no excuse to say the method is "new" Actually it
iS'& method used by Steinhaus and other analysts in 1920-30 before the
notion of "independence" was established. Anyway has it led to any
good results? Since you told me that he was a friend oftoygsso t is
your duty to give him some advice. I will not continue
Rao is coming to the U. S. for one year or longer. If you learn
to give a course in English I can arrange something for you so that you
can report to the American probab ilists the work done by Chinese. .But
we cannot invite people who cannot give lectures in English.



July 24,: 1980

Dear Dr Chen,

It was a pleasant surprise---your letter. A few days ago Chern
asked me about your conference but your invitation to be external exami-
ner was new. I recall our talking about it on that trip to the Pinnacle
My congratulations for your sumess and more success in the future. The
merging of the universities seems (to outsiders) a correct move. Maybe
you can now obtain French funds to further your research program.

I must begin by saying that now that I know the dates of your con-
ference, they conflict directly with an Oberwolfach meeting of which I
am co-chairman. In any case as I told Chern I would .not go to such a
conference unless travel expenses are provided for. Now coming to the
question of external examiner. Iy experience as such in H- Ku was a
mixed one and I would not want to repeat that. However your presence
and the novelty of Singapore, and our previous talk about it inclined
ie to acceptance. Maybe you will solve another problem as you did so
well last time when you were here. Then are some questions I must ask
before a decision. Naturally the interesting thing is a visit to your
country during the term. Reading the specifications about that, I should
like to have a few points clarified. First I would like to make this
visit during a relatively cool period, perhaps around the time of Decemb
xtaxHx say. Will your university be open during that period? Next
it is likely that I want to combine such a long trip.(and economy class
---unlike the great H. K. U.) with another stop or two. Of course I
wsuld not want to cost your university any more than the shortest flight
but I must have the liberty to take a different path. One can do an
arithmetical computation in that case, etc. I am a little afraid of
Asian type of red-tape and therefore must inquire in advance of such a
possibility. By the way I may also combine t-is visit with a trip to
rRoC, in which case the time will be perhaps somewhat earlier such as
November to December. For I have already decided that my next trip to
China will be in the early fall, quite likely next year or the year afte

Having glaneed at your papers and taking such matters quite more
seriously (than for instance Chern would), I must confess some of the
subjects are no longer fqcmiliar to me (though they must be fourty years
ago, such as differential equations). So I should really waive compe-
tence in some of these areas (most people should but probably won't say
so). On the other hand your problems in probability (and some statistic
are within my competence, and I should be glad to be of help.

How much is a Singapore dollar? Is 150 per day sufficient for a
reasonable hotel room (with air conditioning if hot)? If so S. is in-
pensive. I just returned from a trip to six European countries with my
son. The average hotel room (with bath) cost o US dollars and meal
cost 2 for two. Sometimes my son was still hungry after the meals.
Please give our best regards to ydur very charming wife. I am still
sorry that you were in such a hurry when you both came to our house last


4 September 1980

Professor Kai Lai Chung
Department of Mathematics
Stanford University
Stanford, CA 94305

Dear Kai Lai:

Thanks for allowing us the use of your review of the
Princeton proposal. I think it would have been unfortunate not
to have had the benefit of your views in making the final

I've enclosed a copy of the review which was retyped. The
first paragraph, an explanation to Bill Pell concerning your
anxiety about being identified was deleted. In addition, we
removed references to Berkeley, Joe Kohn and Joe Doob. I don't
believe that we can go much further without (legally) negating
the review.

It was good talking with you. Perhaps I'll have a chance to
visit Stanford sometime this year, it's been a long time.

Sincerely yours,

Jon V. Ryff
Program Director
Classical Analysis


Program Director
National Science Foundation
Washington, DC 20550

Dear Dr.

Princeton is an excellent place for mathematics, at least in some areas. It
has produced many good people. There is no question that it can continue
to do so. Therefore it seems to me that the pertinent question to raise about
the present proposal is: does it need such a special project to carry on? I
talked with some people who seem to know the story. Apparently this proposal
is a kind of substitute for an Institute............................Prince-
ton cannot get it perhaps because there is already an Institute in the same
town. So Princeton wants a consolation prize because some of the more
aggressive types ( ) want more money and power. What will
they do? They will invite a few "big shots", make NSF foot half of the bill,
and then appoint a larger number of younger people around these, again let
NSF foot part of the bill. But these things can be done under the present
setting of NSF, through research contracts and grants, and through special
conferences. What Princeton wants is to have more of these for a longer
period, and probably with less control once the money is granted. In short, they
want a bigger cut of the pie. Is it worth it? For certain people, and for
certain areas of research, this bigger support may produce more results. Some
of it will of course be wasted in unnecessary expenses such as travel. Some
"buddyism" cannot be avoided as it is well known that there is a strong mutual
admiration society among the VIP you invite me and I invite you, etc.

With inflation that high, I do not know the latest picture for NSF funding for
mathematics. This will have to be an important consideration for any such
project siphoning away a lot of money. For another reviewer, I suggest ..........
He is relatively conservative on these matters. Incidentally, as far as my
own area of interest is concerned (also......'s) it has been dead for many
years at Princetfon. So we can claim some personal disinterest in the matter.


Dar Dr. .IAling.
: :I am writing you about a use of Vitamin C which'. does not seem
:to' be v" ll. knbown. Namely 'pplied in. it powder form. to canker-sore.s
inethe mouth"or on the lipsa it is a quick .cure -Applied this way,
before, the canker-sore has broken loose it nips it, in the bud. .I
,found .thi, by experirienting with myself some years ago when I was tak-
ing Vitamin t: after reading about your discovery. I,have told this to.
-sdme physician? who' had not know about this use. Until my little dis-
covery : used to earry a bottle. of. Tincture .f :-rrh during travels"
as a cure f6r such so.res. But Vitamnir C powder is a lot
easier to F-t, carry -.nd :apply.

t would be nice .if: ycu could either substantiate or refute my
observation above..


Bismardistr. l'/2
UNIVERSITAT ERLANGEN-NORNBERG D-8520 Erlangen, dmw Sept.25.1980
Pro .Dr. Heinz Bauer Tel. (09131) 852453 (Durc wahl)

Professor K.L.Chung
Stanford University
Department of Mathematics

Stanford, California 94305

Dear Chung:
Many thanks for your letter of July 22, which arrived here rather late
after I had left for vacations.
I find your results presented in the C.R. Note very interesting because
of two reasons:
1) You treat the Schr6dinger equation with the same process as one
uses for the Laplace equation (namely Brownian Motion).
2) You treat for the SchrBdinger equation a boundary value problem
without having a maximum principle (at the boundary) at your dis-
position. This can be seen by looking at the one-dimensional case
(d = 1) for q = 1/2 and by taking for the domain D an open inter-
val of length 2x. The solutions of the differential equations are
then given by
a sin x + b cos x = A sin(x + c).

The relation to axiomatic potential theory becomes clear as soon as
you come to Theorem 4. Indeed, it is known that for continuous q
the Schr6dinger equation leads to a Brelot space (see e.g. Bony,
Annales Fourier, vol. 17.1, and Brelot-Choquet-Deny Seminar, vol.12).
The condition that a strictly positive solution of the Schr6dinger
equation exists, then leads to the fact, that all relatively compact
open subsets are MP-sets in the terminology of Constantinescu-Cornea's
book, that is that the boundary maximum principle holds. When the
positive solution has a strictly positive lower bound on Dthe boun-
dary maximum principle is even valid for D.

I wonder why Brelot did not tell you all this. It is certainly known to
I would very much like to see details and proofs of your result.
Please give my best regards to Murali Rao who as fas as I know is now
in the United States.
Best regards.
Sincerely yours,

October 19, 1980
Dear Lars,
I am inclosing a MS for the Acta Math. It is a great paper
but I am not sure you or anybody in your neighborhood can read it.
I hope you will be able to have it refereed in aebet <,-three months.
You will hear from me again after this period of time.
How was your experience in China? Now I can tell you that the
conference was opposed by many Chinese mathematicians including
Hua Su and others as being "premature". Judging from my own
experience with visitors they need a lot of time just to catch up
with the basic literature in mathematics.
I will be in Europe at the beginning of June next year for a
few weeks. If Sweden is not bankrupt by then I could hop over to
see you but all my expenses must be paid.

Please acknowledge receipt of MS.


U. E. R. 48.00

4, place Jussieu Tour 56
75230 PARIS C6dex 05

Tl. : 336-25-25 )Postes 53-19 et 53-20

Paris, November 14, 1980

Professor Chung
Stanford University
Department of Mathematics
Stanford CAL 94305

Dear Kai Lai,

Since your letter of October 9, I met Paul Andr6 and Jean-Pierre (Meyer
and Kahane...), both back from China on very different expeditions. I expected
news on our project of a Colloquium from them, but they did not essentially
learned anything except that Wang is still very interested by the project (he
mentioned his interest to Kahane).

I am not very enthousiast by the Tien-Tsin project probably because of
all the places we visited together, Tien-Tsin seemed to me the least interesting
from the non-mathematical point of view although it has the advantage of being
close to Pekin.

I feel that you should take the decision between the two proposed solutions
as you have better insights of the situation. I expect no difficulties to get
full travel funds to China for five to six French lecturers, whatever project
we opt for, provided I can present the retained projected mostly as a bilateral
Chinese-French one.

Best regards,


Springer 4
Springer-Verlag r
Berlin Heidelberg New York

Professor Kai Lai Chung
Department of Mathematics
Stanford University
Stanford, California 94305

November 20, 1980
RM/mrk 2141.

Dear Professor Chung,

Thanks very much for your letter of November
translation of Hou Chen-ting's book.

12 about the English

I am disappointed to hear that apparently no notice was taken of
your suggestions. I was also under the impression that Hou was
involved in the translation, so it seems rather surprising. I shall
do my best to elicit a response from them. I shall let you know
what I hear.

We did not receive a reply about Wang Tze-kwen's book.
Science Press about it again.

I shall ask

With best regards,

Yours sincerely,

Roberto Minio

P.S. Dr. GStze is away on business at the present but I shall show
him your letter as soon as he returns.

Springer-Verlag GmbH & Co. KG
Postfach 10 52 80
Neuenheimer LandstraBe 28-30
6900 Heidelberg 1
Pers6nlich haftende Gesellschafterin:
Springer Verwaltungs-GmbH

Telephone: (06221) 487-250 Eingetragen im Handelsregister des
Extension : (0 62 21) *487-1 Amtsgerichts Berlin-Charlottenburg
Telex :04-61723 unter 93 HRB 7812 und des Amtsgerichts
Cables :Springerbuch Heidelberg unter HRA 1007
Dr. Dres. h. c. Heinz G6tze Dr. Konrad F. Springer Dipl.-Kfm. Claus Michaletz

MATHEMATISCHES INSTITUTE Erlangen, & November 25,1980
Postanschrift mailing address:
Prof.Dr. Heinz Bauer Mathematisches Institut
University Erlangen Niirnberg
Bismarckstrasse 112
D-8520 Erlangen
STel. (09131) 85 2 4 5 3 (Durchwahl)

Professor K.L.Chung
Stanford University
Department of Mathematics

Stanford, California 94305


Dear Chung:
Many thanks for your letter of October 27 and above all for the in-
vitation to the Oberwolfach meeting in June 81. I shall be most happy
to participate.
I understand that you have difficulties to find the general version of
the maximum principle in C. C. It is formulated in Theorem 1.3.1.
in a very general form. What is behind it can be found in my Springer
Lecture Notes (Vol. 22) "Harmonische Rdume und ihre Potentialtheorie"
on pages 25 26 and for the Brelot case in Brelot's Tata Notes on
pages 71 72. For your case (the elliptic Schridinger equation)
Brelot's setting is perfectly sufficient.
You ask for the axiomatic way of showing that a harmonic function
(defined by the Gauss arithmetic average property) is finite every-
where when it is finite at a single point. The answer is simple but
probably not satisfactory for you: This property essentially is the
fundamental convergence property of Brelot and in almost all applica-
tions proved via Harnack inequalities. There is a good reason for this
because the convergence axiom and Harnack are equivalent according to
an observation of Mokobodzki. You can find details in my Lecture Notes
on pp. 35 and 175. (Please observe that in Brelot's setting the total
and the empty set are the only absorbing sets.)
I am happy that you sent me your complete manuscript, which I have not
yet studied in great detail, because of lack of time. I certainly will
study it carefully within the next weeks and I may than have some
questions myself. In any case I find it most interesting what you do.
I just wrote a survey article on harmonic spaces which will appear in
"Uberblicke der Mathematik, Jahrbuch 1981" (published by Bibliographi-
sches Institut). I tried to work out the main features from the point
of view of applications. You will receive a copy under separate cover.
Perhaps you are interested.

Sincerely yours,

November 2, 1980
Dear HTsu

Your letter is interesting. You seem to have a lot of energy
to do many things. You like to write, and should be an editor.
The name suggested is not goods the word researchera" should be
"research". The words "and reviews" are not necessary, but if you
wish to emphasize that the journal is not just for "research papers"*
then "Journal of mathematical research and exposition" seems better.
I have a short article of some historical interest which I may write
and send you, but what time is uncertain. Do you want articles in
English? I will ask people like Erdos44, p Ce-e cl.
About."advisory board", donkt you think names like Hwa and Su
are just empty signs? 1S4 Do you need such signs to get
something from the giverhment. Do thy, need such "honor" to help you?
How will they help your journal? By chance to-day the local paper
has a long article about that con man (Li day's "son") who had the
use of Su's black automobile "for as long as he wants". Did Su also
want to marry One of his relatives to him? :ixxyxxxxxiPxxxmxkza
xitaxikxtkAi BK x Are they (incl. Su) ashamed of such doings? You
are an early comrade. You should express your frank opinion. Do you
think science will proper under this kind of atmosphere?
Pan Am will begin a flight to China spzt n from San francisco.
I think I will take it but my plans are3fixed. If DIT can take care
of a.l expenses (incl. train ot airplanes from Beijing to Dalien and
back for a week, I may visit you next September. I can give two or
three lectures at different levels. In this case Ian you also make
arrangements with one of the universities in Nanking/Soochow/Hangchow
to do the same for about a week? I prefer Soochow but am afraid the
"Normal college" is too small or too poor. In anycase I want to spend
a week in that area. Then another in Huangshan (I'll write them direct-
ly but if you know someone well you can also make the contact). If
all these can be arranged I intend to come in September 1981 for a.
total of about one month---a few days only in Beijing. Since you
know lots of i:ople maybe you can arrange this for me. It usually
takes too long to ceal with Chinese, so I am writing this early.
However all the conditions must be met. In the past these things
were left very vague, making it difficult for the visitor to decide.
But I hope now the situation has improved sufficiently to allow for
advance planning. Please write me agian when you have some idea if
the arrangements can be made.
Why is HIT so powerful? Which mathematicians are there? Wuhan
Changsha is too hot in the summer.


Exact dates and other details will be fixed later, but not
later than next spring.( In the U. S. and Europe we have to plan
such trips at least several months ahead of time---something the
inSt ebs shHdng a n er xlnedspinghave an alternative plan which

Nov. 2, 1980
Dear Shizuo,

I have not seen you for a long time. Sometime ago I saw an
article by your daughter (?) in the N. Y. Times Book Review.

I am writing you to ask you a couple of questions regarding your
famous paper "2-dim BM and harmonic functions" (1944), of which I
have a reprint inscribed by you.
I. Did you write a sequel to this paper giving details which are
omitted there, see e. g. line -7 to -3 on p. 709.
II. In formula (15) of your paper you gave the probability solu-
tion to the D. P. To-gether with Theorem 1 it will also give the
convergence to (continuous) boundary function for a Jordan domain.
Unfortunately the proof is omitted loc. cit. Since a Jordan curce
is only continuous without further smoothness (0 for instance) it
is not trivial to prove the convergence as stated in (7). You have
defined the regularity of a boundary point (before Lemma 1) but it
reamins to show that each point on a Jordan arc is regular. (In your
terminology, for each point which is an inner point of an elementary
set E of the boundary.) I don't know how to prove it.
In Doob's later paper (1954) he referred to your results but did
not solve the DP by probability methods. Instead he used a PWB solu-
tion to prove results on stochastic boundary, namely boundary follow-
ing the paths. I asked him about this which led Iro my questions above.
Of course, looking back it is very easy now to prove the convergence
to the boundary function at each regular points an shown in Ito's
Tata lecture notes. This is the first place I saw it. Do you know
an earlier reference?
Please answer these questions immediately as I am writing some his-
torical notes for the material xxtxx~tia. Also, can you send me a
reprint of your talk in AMS on Brownian motion? I don't have it.
With best regards,


BM = Brownian motion
DP = Dirichelt problem

ov. 27. 1980
Dear Prof. .IIsu
*arI; Iplas'ed with your. response ax in your letter of the 18th.'.
Your riew. journal .should -serve : p',rtly the purpose suggested in my.
'letter to. !'iu, a.copy of which is incloed. [Please keep, this discreet
.as I do not wish to case people to lose face unnecessarily.] The
w rd "Ixposition" covers the kind of articles which &.v not acceptable
as research, but still- of value to stutden~s. and teahherse. It isa good
idea to devote half of the Epace to such, articles. ..On the other hands
ahZs' rFtcs with no indication of proofs nor any guarantee of veracity
are ,lzeaess and only .,asteoif time and paper (even hough they are used
to advertise for pro motion'. I understandd. They should be limited to
exc- Flt-onally .interesting announcements and required to contain some
useful':deRc.rir),tiorn~ FThe recent Abstracts sent to me' from Fudan con-
tain: y'.i' t..nrinucp !-nte of no: value at'. all ,"' such as .thbe by Kuo. ]
.Ercdo '.ie oC arourid .t':' yiill. ask hi.m When, I find' ouit Where he is.
I will- alsc ,t soie other good apers for you in ,the'. area of *conmbi'r-a-
toric/analysis/probability,. Ad'viuors. are useful only if they are res-.
ponsible anda 'cnoleda.hle. Frdos is appropriate as a honorary member
(he helped. China: a', lot ain many ':ways). willing to. accept such an:
appointmentn f.or :you.rujournal but i' this casee, iA mut be consulted on-
other appointments. Since you, are chief editor it will be easy for us
to discuss .ic1 "matter: but as you, know among foreigners there are'also
undesirable advisors. () .
i:'Jf.y' :'%a 'nt an article for your first issues the only thing ready.
is my lecture notes on Browniah motion on. the line, given last year
('.i~ bRong aT.tended a few classes) and originally intended as Chapter
I o.f a- small roongrph. So far only< about -O sparsely typed r-ajes are
written, and farther progress, will be delayed. These notes are at,
the level of "exTosition".. with some new proof ( ,.:. it little like.Hardy
and Wriht.s Number Theory which wa.s one of my favorite books, in Kun-
ming). Theye are -oe exercises for the 'reader.. It may be an 'iterest-
ing innovation for your new, journal t'ohave such a "e'rieal" ,. :
to be -rinterd'in two or"three However I must retain all copy
rights and forbid Chinese translation without. my permission.. I am send-
ing 'you ithe first 18 rages under separate cover. You *iedodqt take it,
if it ips not suitable. Another possibility is .toa pubiisfa very small
book either,'in English or in Chinese,-which will be usefull for students.
If vou. can my trip nett year-as'-discussed I should be most
pleased. 'After you have .made contact with I.uangshan asd 1anking/Soochow
S.Ad after Pan Am. announces its new flight's to China I -can give you more
details., Roughly ,speaking I wish to spend part of'September in the.Nort.
and the end of September andearly October in the South.. It is possible
that. I ill -pe.nd a longer time (in-Octo:ber) in China if.I can arrange
it. I have a sabbatical., but part of it has already been planned in
Europe, I do not know where in Australia'.your friend wants-to send his-
calli.gtiphy. It is very :important to .have the address .and dtaes.
If he .knows these I shall be glad to send it on his behalf and corres-
pond wih -the- people in Australia.- With best pregards,


Dec. 9, 1980
Dear Falkner,

Chacon was here and we talked about having you come here next
summer. If you will be on an NSF project then there should be no
objection to your spending sometime at 'Stanford. If you are apply-
ing for such support you may consider- this possibility. I shall be
away during part of the summer so in case you want to came we should
fix some dates. 71e can offer some supplements to transportation and/or
living. Unfortunately due to the lost letter I could not put you on
my project. Let me know soon what you plan to do.
I believe I Fent you our C. R. note. The complete MS is now avail-
able. 'There are sc veral directions 1 should like to extend the work.
For instance, the general elliptic case., or een more general processes.
Theorems 1 and ? for unbounded domains 'are still open. Ahalytic condi-
tions for the fintteness of u Comparison with eigenvalue problems.
H. Bauer rointod out some connections with the axiomatic theory. In
Brelot's case, apparently only the (trivial) case of nonpositive q
was included in the axiomatic. Now 'e know it can be extended to
a bounded q, etc.




Ad 03

Dec. 28, 1980

Dear Falkners
I am pleased to know that .you can come next summer. Why donXt
you come toward the end of June and stay as long as you wish? So
far as funds are concerned the manager in the department told me that
we could.arrange to pay you one thousand dollars to cover all. Even
this sum was obtained through some juggling. so I hope you will accept
it a"s a modest contribution to your moving here for a while. Upon
your formal acceptance I will get you get ii touch with my secretary
so that you might get into university housing which is a lot better
for you. I must say that I shall be in Europe until about June 23
and there is a probability <1/2 that I will be away again later in
the summer. However it is, certain that I will be here for more than
one month from Buly to August, not counting September during which
my plans are more indefinite, It should allow us enough time to go
over ,oine -f the probler-iBs you may want to work on If so I will try
to add you to my project for the following summer.
I ami sending you a copy of the MS. Pease send any comments. Re
your question, iy is a folklore in PDE that the decay'rate of Qt1
must be more rapid than some exp(bt), 6>0, or growth
else it will converge to zero as shown in our MS. This has to do
with the first eigenvalue of the operator. The finiteness of u, is
equivalent td. "all eigenvalues are < 0", as you can probably prove
yourself. I am writing a short note on these connections with PDE
teo 6* and hope we can enlarge upon it. If uD is oo, can there be a solution
of the Schrodinger equation with given continuous boudary function?
think yes (from PDE) but do not know how to/ treat such problems by
our methods. (D bounded). O 4 0 c e au .etr
Another problem occurred to me recently. Suppose that f is not
bounded but rerely.L on 5D, what can we say about the convergence
at .'D? Eevn for the Laplace equation I was unable to find a precise
answer. .It is related to resolutivity and probably solved by Brelot.
Do you know the exact theorem regarding the convergence as x tends to
z on 5D? Rao said that it was not in Helms' book but I think Doob has
results; on this problem. As you see there are so many good problems
in this. aea even if we suppose D to be bounded,, 'hat I'd like to do
is to explore some of the deeper connections with eigenvalue problems.
Let me ,rrenmark that according to the Alternative' Theorem for elliptic
p.d.e.i if the O-bdy value has no solution 'i..e. 0 is not an eigen-
value), then every continuous bdy value problem has a unique solution
(let 6D be regular). But out result sasa if' uD true,,and uD is the solution of the 1-bdy value problem. That seems
odd to me. If you have questions regarding your visit please phone
me a, home (415)-857-iW 9374 in the morning (my local time). I assume
you have US immigration status otherwise we have t.WYWB'v n6VS6apay
you. .
I havy had no timuO.P flygg po ^.LSent me.Since I have never
worked on that pro U:aH~)d33I fl1 QI~3)ldg~ng to say.


-December 29, 1980

Dear Dr. Gotze,

First of all, my wife and I enjoyed your very pretty greeting
card and wish you and your wife a very Happy New Year. I am sorry
to learn that you will be'too busy to come to San Francisco soon.
I hope to meet your editors mentioned in your previous letter though
at the moment I am not sure I will attend the meeting.

I am writing you directly about another suggestion. You may re-
call.our discussion about publishing a book,by the late Hsu Pao-lu.
There was a formal memorial for him at Beijing University last sep- .-..
member, and a volume of Selected Works by him will be published in
China. The people at Beijing University would like to.see his com-
plete works published in.English abroad. As you know together with
Chern and Hua^you know personally, Hsu was regarded as the third out-
standing, man during the epoch when I was a student in Kunming, China.
His status can be appreciated by the Obituary and three articles de-,
voted to his works which appeared in the Annals of Mathematical Stals-
-tics in 1979, two of which written by Americans. A list of his .publi--
cations are inclosed. The material we talked about is included in Nos.
5, 67 38 and 39. The articles in English can be reprinted. Those
in Chinese will be translated by a small. team organic. ed by me. Beijing
Univer it:y will pay for the translations. The complete MS will -be
edited and deliverable in typescripts and photo-copies to the publish-
er. hope all this can be done within one-year. .Na-turally all of
us would be pleased to have your Verlag as the publisher. Would you
be interested? If so, your editor can estimate the size of the book
etc. A format like Wolfowitz's Selected works would be-fine, I think.
The royalties will be paid in normal manner to Beijing University. As
soon as a contract is signed the translation will begin. Of course
you may wish to consult .ith your staff and perhaps others before mak-
ing a decision. iease favor me with a reply then. If there are fur-
ther details about the publication which we should discuss, please let
me know.
':ith Season's Best Greetings from my'wife and myself,

Yours sincerely,