Barry Simon, 1980-1994


Material Information

Barry Simon, 1980-1994
Physical Description:
Simon, Barry
Chung, Kai Lai
Physical Location:
Box: 1
Folder: Barry Simon, 1980-1994


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

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Source Institution:
University of Florida
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All applicable rights reserved by the source institution and holding location.
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Full Text

April 28. 1980
Dear Simon*

Thank tou for the letter. I reply now because there is still one
date left (June 5 Thursday) for which I can invite a colloquium speaker
and pay some expenses (same max as before). Apparently Carmoha cannot
make its if you or one of your colleagues can* and can tell me something
about Feynman*-Kac stuff of 'interest to probabilists. let me know at
once before it is filled. Operator theory is not of interest.

I have bben preoccupied with finishing a book so have little time
to return to that C. R. Note you saw. I have been aware of the Russian
paper you mentioned, but it deals with the case where q in positive$
so from our point of view fairly trivial. [Picard's method works there
as he showed.] Besides, 3relot wrote that he had done the potential
theory in that case back in the thirties. The case q is negative is
of course even more trivial. From the probabilistie point of view the
case of a bounded q of arbitrary signs is intriguing because the BM does
a lot of the cancellations, kind of unexpectedly. If one puts a condi-
tion (L or else) to essentially suppress either the + or the part then
there is not much to do f By the way ask Carmona to show you the 1-dim
case which I did several years ago. Although the high dim. case is a
lot harder the basic ideasare the same. You said in your letter that
you are interested in q, can you specify the kind of con-
ditions? On the other sile, the basic results in our Note)is only a TOI
little easier even if th*; domain there is a ball! In fact some of us
tried to do that case fi'pt without success. The Harnack part is very
easy for us but extenti6n to non-BM xaet4e would be hard because we use
zaxzagiy spherical ,symmetry. Indeed one interesting problem for us is
what kind of "reasonable" conditions can take the place of sph. symma
but of course reasaonatness for probability is not the same as for ana-
lysis. We want, a "process".

I was, told tha{ the eigenvalue problem is trivial when q is positive*
but he i-cn't 'ay what happens when q is +. I am interested in the connec
tions, probably a kind/ of variational meThod.



P. S. I you hlve a bright student who can tell me things of interset.
I mp(y be able to give hia some. support this summer (qua graduate student)
P. S. 'C3n you answer t@s as physicist? The equilibrium distribution
in electro'statics is supposed to be a steady states namely the limit dis
tribution pf some (ergodic) average. Can you identify the latter mathe-
matically?' Namely find the process of which the distribution is the in-
variant me~asure. Nobody has been able to tell me. I have an answer.


May 27, 1980

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

Dear Prof. Chung,

Please forgive my tardy response to your letter of April 28 but
it arrived while I was away at a conference; I then delayed responding.
I talked to Carmona who was going to call you and then finally there
were delays due to the vacation of my secretary and to the birth of
a child (which would have prevented my accepting your kind invitation
to talk on June 5).

I hope that you will have heard from Carmona some of the details
of the approach that Aizenman and I have been using in our study of
Harnack's inequality.

As I will be visiting Cal Tech for the year next year without
teaching responsibilities, it may be possible for me to visit you in
Stanford to give you more details if that is desired.

As for the kind of local singularities that one wants, one would
like to consider an operator in three N dimensions where a point is
thought of as tA three-dimensional vectors and one would like to handle
potentials which had Coulomb singularities in the difference of some of
these vectors. This is, of course, the case where one does not need to
use cancellations between the positive and negative parts and so it is
a case that I gather you regard as "easy". From our point of view it's
not very hard and if one does not allow the possibility of cancellations
Aizenman and I seem to have found necessary and sufficient conditions
for a strengthened Harnack inequality to hold.

We also discovered an example where there are lots of cancellations
between the positive and negative part although we demonstrated these
cancellations by using operator theoretic ideas. That there can be such
severe cancellations has been known within the field for about 10 years;
see for example the discussion on pages 156ff in Volume III of Reed-
Simon. This deals mainly with strong oscillations at infinity but as
described in the notes on page 361-362, there have been results also
on the effect of severe oscillations cancelling out at local points.

Thank you for the offer of possibly supporting a graduate
student for the summer but none of my current students are working
on probabalistic things.

I was unable to understand what you were trying to ask in the
last paragraph of your letter and in the second P.S. so I will not
try to answer that.

Finally, with regard to the attitude in the first paragraph
of your letter,not being a probabalist myself, I am not sure what is
considered "interesting to probabalists".

With best wishes,

" Barry Simon


May 28, 1994

Dear Barry:
I hope you will get this at home and reply soon* certainly before
June 10 if you want an impact. We are finishing the galley of "From B. M.
to S.'s E." ai*d I [not we] am adding a few touches to my historical ITotes/
[which EVRY3CDY loves to read].
I recall in ycur joint paper* which you completed after you came to my office
and were given a preprint of my paper with liao [remember what you averred
previously about the DBVP?] you criticized us [I shall use the gentler word
"lamented" OK?] because there was NO mention of your key condition (1.11)?
As a conscientious historian I intend to give a delayed reply for the bene-
fit of unreformed analysts who might read the book, to the effect that you
gents wore quite mistaken and did not realize that my [sic] gaugeability
is a sufficient condition, without any eigenjunk. I do not think you under-
stood it at that time* did you? Anyway now we even proved the equivalence
of gaugeability with what we now write as "Xl the way, Falkner informed ne that the explicit determination of X1 as in
your (1.11) for J [your silly Kd] is far from routine, not to be culled e. g.
from Hilbert-C.urant or any textbook. Right? 2ut a detailed proof is now
given. Since your joint paper is famous and read by many analyses who don't
know beans about B. Li. I think I owe it to them teo ay asc a denouement.
But Zhao being a newcomer to theseiaes was scared that my remark might
offend you* although I shall uay explicitly.all the opinions expressed are
mine alone/ I am not afraid of offending anybody so long I tell the truth,
not "hAng frcongressUmH or dogcatber in Hebron. But since I am a nice guy
I a). giving you an opportunity to recant or whatever, e. g. if you ask for
it I can make the authors anonymous, as I did in the Expositione article
sent to you long ago. W~ do you say' Don't mince words)
By the way I praise as well as condemn: I call your equivalence theorem -wj
remarkable and you may even be pleased to see my expose of it. Ve also gave
a totally new proof of your nice result about the continuity of q-harmonic
fucntions, and use zero LP estimates, ha-ha. "True analysis L4 -4.h LI
and L1; when one fails W4SMM these one chickens out to LP."t4--do you know
who said that? I will tell you' if you did not know. Kind regards,

P.S. My e-wol es lost w. or. L.



June 1, 1994

Professor K.L. Chung
903 Lathrop Drive
Stanford, CA 94305

Dear Kai Lai:
I wouldn't worry about offending me if you tell the history as you understand it, I have
no problem.
I may be sounding like a witness before a congressional committee, but I must say that my
memory is unclear on exactly what I knew when. I certainly didn't think of things from the
perspective of what you called "gaugeability" but I would certainly not have been surprised by a
theorem that said that the finiteness of a certain expectation based on stopping times implies a
positivity result of the type of my condition (1.11).
While I certainly do not believe that any textbook dealt with the space that you insist on
calling J, given that J potentials are form-bounded, it is really a simple exercise given what you
might find in Kato's book or in the right parts of Reed-Simon to go from the determination of the
lowest eigenvalue for smooth potentials to ones in class J. (I'm, of course, only guessing what
determination you have in mind.)
I'm unaware who made the quote on true analysis but I find it most amusing.

With best wishes,

/ Barry Simon