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STANFORD UNIVERSITY STANFORD, CALIFORNIA 94305 DEPARTMENT OF MATHEMATICS December 5, 1978 To whom it may concern: Di. Joseph Glover worked with me for a quarter before he got his Ph. D. from U. C. San Diego. He read a MS of mine and added several good results to it which became a joint paper, to appear. The best of his contributions there is: for a left continuous moderate/ Markov proteasy the set of points of any Borel set A which is not regular for A can be met by the paths only in a finite set in each finite interval. ?or Hunt process this is Hunt's theorem but the ex tension to the left case without further condition is interesting be cause there is no zeroorone law. His proof shows adroitness in us ing elaborate techniques. In the same vein he was able to give an al gebraic proof that any predictable time can be announced by countably valued predictable times. This result had been proved before using a measuretheoretic argument which required the discarding of a null set, see the book by iDellacherie and Meyer. According to iqeyer the pure proof may be new. While he was here he participated in mny semi nar and learned solme classical potential theory which apparently was not taught it San Diego. After learning the recent results by Kanda and K. M. Rao on polar sets and energy principle he obtained a result along their lines dealing with the maximum principle for potentials of measures. These are hard problems and his ability to get into the subject shows his drive ind capability. He knows well the modern stuff in general Markov processes a la Strasbourg school and is willing and able to learn other stuff. If he can consolidate and deepen his know ledge there'is excellent chance for him to develop into an accomplished probabilist. He has a very agreeable personality a4ac communicate wells and is energetic enough to be useful all around. I can recommend him highly for a position in any good department. Professor Kai Lai Chung COPY THE UNIVERSITY OF ROCHESTER DEPARTMENT OF MATHEMATICS MATHEMATICAL SCIENCES BUILDING ROCHESTER, NEW YORK 14627 Phone 7162754411 April 28, 1980 Dear Kai Lai, I can't answer all of your questions, but I can answer all but one, I think. Let me give a series of examples first to illustrate various possibilities. Let me first define notation. If A is a Borel set, I'll let TA be the first hitting time of A by the process X(t), and I'll let SA be the first hitting time of A by the process X(t). EXAMPLE 1. Here I give a set A so that rA = Ar, but TA SA a.s. P for some y in the state space (here I mean rA is the set of points x so that P (SA=0)=l ). The state space of the process is a 1 0 {l}L)[0,c). The process starting in the interval [0,) moves to the right with uniform speed. The process starting at the point 1 sits there for an exponential time and then jumps to the point 0 from which it moves to the right with uniform speed (I apologize for the triviality and artificiality of the examples, but at least they illustate the points). Let A = {}1. Then rA = Ar = 0 and P (TA<0 & SA = m) = 1. So this shows one can have rA=Ar without (B) holding. r r EXAMPLE 2. In Example 1, both A and A are empty, and you asked for an example where xEA but not in rA. First, let me point out that the example you proposed cannot happen. Recall that the picture you drew looked something like this: the process hits 0 only countably often and immediately mwe away. The question is whether T =0 under the measure P This cannot be (at least if hypothesis (L) holds for the process). Because then, by Dellacherie's Theorem, 0 must be semipolar. But it must then in fact be totally thin, so the times the process visits 0 cannot accumulate. Therefore, TO > 0 a.s. So let me propose an example. I'll first propose an example which is not Hunt, but which is obviously strong Markov, and then I'll show how to modify it to make it 1 1 a Hunt process. Consider the state space E* = '[0, 2 )x{n}, and let E = r n O,0}ILE*. E will be the state space. The picture looks like this: The state space consists of horizontal lines and the point (0,0). Starting on any of the horizontal lines, the process 4 moves to the right with uniform speed. When it reaches the* end of that line it jumps up to ( ) the left endpoint of the line above it and moves right with uniform speed, etc. ad infinitum. Starting from the point 0, it immediately leaves(0,0) and starts the procedure described above. It is easy to see by reversing any path that one can define a right continuous strong Markov process starting from (0,0) (one needs to use the fact that I 1/n2 < m .) Let A = (0,1/n)l Then Ar = (0,0), but A = 0. Now you may object that the process is not Hunt. One can modify the process slightly so that it becomes Hunt in the following way. I shall describe it heuristically; it would be somewhat tedious (but possible) to write down explicitly the transition semigroup. Let E = (0,0) U) [0,o)x{l/n}. Starting on 1 a horizontal line at height /n, the process moves to the right with speed 1 for an exponential length of time with mean, 1/n2, then jumps to the left hand endpoint of the line above. Once again, this process can be extended to(0,0). Take A as above. This process is now also a Hunt process. EXAMPLE 3. Question: Can one find open sets G containing A so that SGn increases to SA? In general, the answer is no. Consider the process described in example 1, and let A =0 If G(n) is any sequence of open sets decreasing to A, SGn will remain bounded and converge to a finite Gn l under P 1 limit, while SA = 00. As for approximating by SK, I can neither see a proof nor a counter example. The capacity arguments used to prove the TK result in Blumenthal and Getoor seem to me to break down here, so I'm afraid at the moment I cannot offer any help on that one. The purpose of Azema's proof was to show (B) without using (L). I have enclosed a copy of the proof for your reading pleasure. Yes, I am going to France in September, and will be there (in Grenoble) at least through the beginning of the following June. Rochester turned out to be a fairly pleasant place. It has many delightful aspects. Of course, as anywhere, there are some things which annoy me, but fortunately, the pleasant aspects have so far outweighed any unpleasantness. I will probably be in La Jolla at least part of this summer. Regards, STANFORD UNIVERSITY STANFORD, CALIFORNIA 94305 DEPARTMENT OF MATHEMATICS 2/18/1981 Dear Glover, Pleased to get your card and MS. Please ask Bernard to formalize his invitationin due time. It will be nice to have a ride in the country on way to Grenoble. We have to fix some dates though in time for me to decide where to fly back. Maybe Geneva or Paris. Can you do something for me? Mural goofed again this time badly. He blamed it all on the editorial staff of Ann. Inst. Fourier but me thinks not. Anyway please give or send the inclosure to the proper place (in Grenoble) and request in my behalf to have this correction printed in the next issue. The mistake was discovered and corrected by me then sent t Murali to incorporate in the galley. But nothing happened. Also some obvious misprints such as the last item in the inclosure (EVEN you saw it!) found y.y me are not corrected. Please ask the person in charge whether they did receive the inclosure be fore the artdile was printed?? If you send the inclosure then be sure to telephone and get an acknowlegement that it will be printed. Let me know how you make out on this matter? Ask Bernard to help if need beypur French may not be adequate. I have invited MIftro but I have my douts. She talks in generalities. I must have told you the following but let me repeat it to you and may be you can forward it to her too. HuntXs Hyp.. (H) that semipolar is polar is a hard proposition in any setting. Ak known proofs use some reversing, the oldest due to Doob by reversing w. r. to M (lebesLue) and in each [Ot]. Other proofs depend on the continuity of excessive functions composed with paths so that's also a reversing. But so far as I knVn nobody has done it directly by lbeking at the reverse process. Last year I .as told in Zurich that Dynkin had given a talk in which he did something like this in a special case but Rao said it was old hat. The line between old and new is of course somewhat subjective there. rut guys like you, Iitro had spent their lives constructing all kinds of duals etc. So it seems to me that you ought to be able to produce such a.proof even if it amounts to a pret ing p of old hat. Can you? Anything new to report? Regards to bernard, love to Ann. Sincerely, I bet neither of you know an example of aBorel measurable frocesss X(tw) such that a ap w SX(s,.)ds does not belong to Bt There exist examples of X which are not progressively measurable (much better than the one in Dellacherie which is notkosher), but they do not heve the above negative property. COPY 6/,4e__A irconditioned ILMORE B IL TR MADISON AVENUE AT 43RD STREET FRANK W.REGANPRESIDENT DAVID J. ARTIN,M* NAWAER Nov. 2 TEL 9~3 MURRAY HILL 77000 Joe : ADJOINING GRAND CENTRAL Joe: TELETYPE : NY I 3494 CABLE: BILTMORE,NEWYORK How is the chairing? Are you going to invite me? Rick has. Shame on you especially on Rao, to make a totally gratui tous error on p. 196 of your triple paper in the Prob. Th. Etc. lines 1 to 3. FACT ONE: I initiated these studies, before I knew Murali. Read my paper in Seminaire de Prob. XIV (1978/79), pp. 3556 for the history. I hired him.after this. FACT TWO: Have you ever even glanced at your own ref. No. 6 (1981) Shame on you three. Where did I [sic] say it was for R1? Read the results there. YOU should also find the first occurrence of the name "Gauge" which was of course due to me. It is really incredible that Mvurali, one of the three authors, should think 1 our gauge theorem was proved only for R BOTH facts are in print. Check them. I am asking you (singular case) to send in at once a correction to Kallenberg to be inserted in the nextappearing issue. If these authors were people we do notknow [I donit know e. g. who Balnchard] I would let it go, but not when one of them is Rao who is also coauthor of two of your referenved papers. There is a limit. By the way, I advise you to write the Correction by yourself, a few lines suffice and I have given above all the printed facts. You might remember that I asked you to go to the Fourier Institut to insert an Erratum for a paper by me and Rao? They duly published it but it was ALL WRONG because in a mo ment of weakness I let Rao make the correction! DonXt you think you should know 'better now? If I were chairman he would get no raise this year PLEASE DON'T SHOW HIM THIS LETTERI! Await your quick reply, and best wishes, "MAKE NEW YORK YOUR VACATION CITYAND THE BILTMORE YOUR HOTEL" e June 2, 1996 Joe: C acauv Have you seen the inclosed? It's been out 'months. Rave letters from all over. As a bait I will give you a copy (free of all charges) provided now you "whisper" (by voice if you wish, to avoid o ,toe but I will not divulge MY source) the neme of that anonym (in my LevY article, printed twice once with the Postscript in BUR Seminar) that Azema, if not dothere has told you. As a lure, I will then send you a copy of my recent communioationwih ulAndre on the affair. A sequel to the Green, Brownu (, sic) .. will,recount the wonderful (fren story with names like Meyer, Neveu, Schwartz, Merdes Sal d, Azema (ha!) and one more mystery entry to be revealed in it, FOR THE FIRST TIME. Do it now* don't wait. /X Oy CL Your sketch was appreciated. When will you make an appearance here again? I have other good stories to tell. Have you written your REVERS" ING? /(0? m2Ar te <0, Sjinoerely, P. S. Reply by mail no e or f imerde). SCEEAR: YOU have no responsility to cover the Frechy arsee or shit. C4te^U^ 