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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 1dim 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 not.soregular 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 nonBM 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 icn't 'ay what happens when q is +. I am interested in the connec tions, probably a kind/ of variational meThod. Sincerely* K/ 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. Princeton University DEPARTMENT OF PHYSICS: JOSEPH HENRY LABORATORIES JADWIN HALL POST OFFICE BOX 708 PRINCETON, NEW JERSEY 08544 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 threedimensional 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 361362, 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 BS:lb 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 your (1.11) for J [your silly Kd] is far from routine, not to be culled e. g. from HilbertC.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 qharmonic fucntions, and use zero LP estimates, haha. "True analysis L4 4.h LI and L1; when one fails W4SMM these one chickens out to LP."t4do you know who said that? I will tell you' if you did not know. Kind regards, P.S. My ewol es lost w. or. L. CALIFORNIA INSTITUTE OF TECHNOLOGY ALFRED P. SLOAN LABORATORY OF MATHEMATICS AND PHYSICS MATHEMATICS 25337 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 formbounded, it is really a simple exercise given what you might find in Kato's book or in the right parts of ReedSimon 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 BS/cg PASADENA, CALIFORNIA 91125 TELEPHONE (818) 3954335 