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5.s(b we obtain (5.15)! except.... Except that Doob's result, Corollary to Theorem 4, is NOT applic able. It applies tp y(Qt+l), not y(Qt). That puny "1" makes a world of difference, and it is t*hi$~iscussion of RANDOM TIME is all about. A Even though Qt is optional, we do not know if y(Qt) be haves like y(l) or y(t); indeed we don't know if E(y(Qt)) is finite! In the very special case where the common distribution is the exponential men tioned above (see below), this problem is a famous PARADOXe, pes redoubtable Feller e~ae h ioutj. see [P]. In that case however we can discover the true distribution of y(Qt) and therewith prove that it has a finite mean. In the general case we are discussing the re sult seems unknown and so the argument above is invalid. Of courses if we add a tiny extra hypothesis, for instance if the y's are domi nated by some big Y with E(Y) mathematician, unlike the practicing scientist, tends to be perfection ist. Prof. Wafsh, Math UBC ee ) 9 [se belW the foSt (aous Changes to make in your NOTES SLbP OCoS Cf by 164: though Fller. 'Y1V skhoL\, citeuit REA) 164, line 2: proofs hy ANOQas nY my M.C. 165: CITE and QUOTE (in his words) the page I faxed you months ago about the lack of stationajty of revers. J'eh insite. He was honest and dead. Many probabilists did not know that, you were too young to know. Aren't there too many Walshts referred to? Of. Part 1i Sec. 6: No agasa and probably both learned from Hunt's discrete t#me chains reversed from last exit times. I reversed CONT. TIME chain (READ my M. C. book, 2nd ed.) T the first infinity TIME b of the "minimal" subprocess that worried Peller and Doob so much SDoob's ghostlike .turn to earth was viewed (before your time) with the utmost alarm and mistruct. All that history was men C tioned in my notes in M. C. loc. cit. Quote it if you can. < Nagasawa did well but certainly not before moi. Did you put (is my reprint on Martin bdy in your Ref? T~Inot in mine. DoAc~cSf Shtryndforms: did you show an importantest case where the h is a Martin bdy harmonic??? If not, try to add it as a suoerb examp le and refer to the Br. Motion case in ChungZhao X(FIND and SREAD it, in a Lip domain). It is this kind of concrete cases That most/ readers should be taught, not the general stuff. Namely,that htransform menas conditioned on X(T)  did you know?? I wonder if Doob did. PLt 1n iYur Pef. Sreduite: How about interior one? Brelot (with whom Doob meant to do o a book together) and some other "frogs" should be mentioned if you know what they did. I invited Brelot here to give a course but I forgot : there is his books) or ask somebody who knows. Holes: your coauthor cannot begin to understand waht you say there: redo it much more clearly wj proQtAesi In case you got my fax ith Inq: you will mail her the entire MS with all of my original text, Preface (one error corrected) AND REFs INCL. You do your Refs. The INDEX should be combined: just add your new itemS I sent you my chosen TITLE, and my part of the BEW preface. Am waiting. r cREAD the first page + par paper and check the dats of the d: ; references: Hunt 1960, Chung [4] 1196%. Japs 1964. Rewrite or just /yc copy 9 whaj I wrote therefor SORRECT history. Pleass add my [4] in our paper to your Refs  I,unlike another author did not put it in my Ref. [Both will be in new book(] perhaps because Chains are not in the Lectures, now they are in the new book. Prof. Wa/ah. Math UBCt ( tIe. beloa r.k most Afhmous Changes to make in your NOTES SqvLbproeVs PoWptat by, 164: though Ffer, .y f ctet RE6AZy 164, line 2: proofs a y h s e n my M.C.I 165: CITE and QUOTE (in his words) the page I faxed you months ago about the lack of stationary of reverse. J'eb inside. He was honest and dead. Many probabilists did not know that, you were too young to know. Aren't there too many Walshts referred to? Cf. Part 1. Sec. 6: Noagasa a~nd probably both learned from Hunt's discrete Stme chains reversed from last exit times.* I reversed CONT. TIME 7 chain (READ my M. C. book, 2nd ed.) To, the first infinity TIME . of the "minimal" subproceaesthat worried Peller and Doob so much f Doob's ghostlike uturn to earth was viewed (before your time) ' with the utmost alarm and mistruct. All that history was men S ttioned in my notes in M. C. loc. cit. Quote it if you can. Nagasawa did well but certainly not before moi. Did you put s my reprint on Martin bdy in your Ref? Titnot in mine. Doafdw A htrandforms: did you show an importantest case where the h is a SMartin bdy harmonic??? If not; try to add it as a superb examp lQ Ie and refer to the Br. Motion case in Chung4Zhao (PIND and READ its in a Lip domain It is this kind of concrete cases That most/ readers should be taught not the general stuff. < Namely,that htransform menas conditioned on X(T)  did you know?? I wonder if Doob did. .4 aut I fYouy Qfe j Sreduite: How about interior one? Brelot (wi+.h v hom Doob meant to do a book together) and some other "frogs'" should be mentioned if you know what they did. I invited Brelot here to give a course but I forgot : there is his books) or ask somebody who knows. Holes: your coauthor cannot begin to understand waht you say there: redo it much more clearly w: r pra+,Aes, In case you got my fax with InQ: you will mail her the entire MS with all of my original text, Preface (one error corrected) AND REPs INCL. You do your Refa. The INDEX should be combined: just add your new item I sent you my chosen TITLE, and my part of the BEW preface. Am waiting. READ the first page p ptr paper and check the dates of the references: Hunt 1960, Chung [4] 11960, Japs 1964. Rewrite or just Scopy O f whaj I wrote therefor eORRECT history. Pleas.a add my [4 in Y0441 e our paper to your Refs  I,unlike another author, did not pu" it in my Ref. [Both will be in new book(] perhaps because Chains are not in the Lectures, now they are in the new book. COLE POLYTECHNIQUE FEDERAL DE LAUSANNE EIDG. TECHNISCHE HOCHSCHULE LAUSANNE POLITECNICO FEDERAL DI LOSANNA D6partement de math6matiques 1007 Lausanne 26, av. de Cour T66lphone (021) 277591 e, c^e OA. L 0 "T ^ L M. J co J ~ i n ^ tr T ^ J C J ^ ^ u z e c lo ~a ^ 4" 6 1^ GL &A, gcJ A Ao 'C^ rQu^i^L~CtLl~ eb>^^J e^OC v^LA^^,' J)de rvf7L, (^ A~eL *J2rC .All ~^^ C^pL /<'< LOdA<^of bi 1o<^a 7 tlet y' ^CA p ) i ^ et ^ ^ A J. AJ '^,> . ^x deAc,A L^ T^tiA a 7^^, ^^.y Ihce an><>jLA ^P~ ^ ^^U u <^)<^ ^6 ^^fuL p se;Wc^ _eee ^^ aS5c ^^< u^ ^^~ ^ ^ ^^. 3g^ /zc ^^ " k.,, r t+. Dr .. a' olt bAIL~J ACw~  2 , c2  '.^iA + /w2 oouI& 0n4 TtQt y^Cl .^. #A L tLtuOJL 1L4h i4f s\ je{ 'J /M~v^ J p, oac \ Qy ~r AAf ,P 4e,+ /i t s wc^ C fi f Y) Sqf(f( 0 f (y)0 ) = 3 .f(X) w ) : f /f' 2 f{ ) 4 ) + C ^,uZ . [o y sa l^ Aw ^^ ^ A CA f2Io ftA dt t~d ~P2~,3~_ SrP O,4td 6;7 4f ~ ~ THE UNIVERSITY OF BRITISH COLUMBIA VANCOUVER 8, CANADA DEPARTMENT OF MATHEMATICS 30 o  D i tCdLZ Lc. TaA aA* 1^ Lw.6 4 Jt 40.1 d 4AQr 4 4 l st5. t ,,44L. 0, ,.' i O '; ~, I, J CLrI Ik1 I u^ oU~ / e^ ~t<^ itZk Ms~s.. */o 2 A 3 ( LCa wJ ~A~oj~ 4 ~I~4~# A *J (A ? f tiM 0 9," 6 AA., w ttt ?AU z E .t aJ MtAtA t< L pIA ( I LaLd~ .L A o IP f sJ o Si. ;^ 4qa q . 4t4sAj .0. So kc, ko if 16.. 4 L;AL 7L 'fr ta,') s437 t a 4 a. AI A l cc l Liz A^LA^ "r B^< 4!!T. <_ C JL. rf YVM J^. S.8q;U% A +^ :t ^^ > IALJ 4oo. THE UNIVERSITY OF BRITISH COLUMBIA VANCOUVER 8, CANADA DEPARTMENT OF MATHEMATICS 4Dea7 lesO La1 . 9A K LK I c's tJ d k 100. b 4, faJL, LA 4 cb fit fJg.9Iwz .I to k iu ^* ^ ;t p0, t X d l dV 0 o kM 4 A e" .Ui AIk tt p' l# dA. AUf^ kio di '1k :,l. 3 X Pdr*.AAk. 1 u i & 0wtA, AEL J C3 c.A A T1 ^^4 ioux ^v\UicJ*. 4 iJii4 *AOti~ tL4*o: rf X ,iA *Aa. j kJ c /uwr & IL > < ) ' X4 X r) > ,+ '. L4A 4  I rUL X k (os. J4 S, vb J r)^vZAl 1ojLJLA o.A.JJ L, x l OS. "( ?5 o L AJ a (n ) ; Tb 1T" A 1 +.^ 4 ,,rA ii' .' A SO dLL* t. LI) u a  1GA% ^ 7 ^ S 4a rA4> TT ~ Yt c l t 7 r= >o (0 4T  SL < T, 7I A 4 7T seLX, ',^ ya 1 4 Xt e 1 / . v uk 1. ta6 Tt> co >^ cy A 0a. so ^ L as "w SJ f ( Aty ^a, w  Dec. 11. 1996 Dear John: Your were such a good letterwriter that I thought I would try again, with several disparq requested (1) At a recent reception a young^,aeme wh name I did not catch (He...) told me that you had proved something with two renexl sequ ences. Can you send a preprint? t the ErddsPolladPeller theorem. I looked at it but forget whether the case Q is special  if soa no good. A few youngsters have also given a "new" proof using coupling, but theirs definitely made the Cb case more difficult  therefore not worth it. As you wall know, in the Erd's proof 0 is the easier case If you did not know, read what Landau said about Erdbv's modf/p a0 theorem. I saw Halberstamwthe rec ttjiea topand he seemed to remember.X I also quizzed him about Hardy e' overlookiAg the genius of Erd8s  the only "excuse being Hardy was quite old already when he met the young E. Surely E. was in the l Al sse Ramarujans and probably in his total output surpassed the In dian. Agree? (3) What I mow seek is a brand new way of the EP theorem with out special argument for the 0 case. It does not exit. (4) Did that Chinese convince you that his proof of that David Pn result is correct? But I am sure nobody knows how short it can be made. Read Dyson's brilliant short proof of the a1 theorem. (5) I will publish a persoril obituary for Paul Erdba. I am also writing an article : "Probabilty And Doob" which intervened un expectedly I O+' year /41~f e (6) I believe you took me to the high table at (xfor4 Jesus Col lege/ Anyway do yoL know Womersley at that colleagues His little gpa4 book in the miniPenbuin series sL know my GibblnRs but that is a snippet from the tome] contained a fantastic mistake. I should like to en TM lighten~B(and also ask another question re Gibbon) but knowing some of those Britishers I should be vexed not to receive a response. If you can make him bahave properly I shall send my communications) via Syou. Let me know soon. Best wishes for Happy New Year! P. S. I do not do computer but do receive email (too frequently) with probadlity 1e C[Littlewoddi 53. but for any important matter a letter salould be sent simultaneously, cI (7) for a few hours I thought the Austin proof of ury the prettiest heorem ai'n ALL Markov chain theory can be somewhat made more intuive by not invoking Lebesgue's monotone function theorem Cin your as well my bool s but it failed. If you are qssSL interest ed let me know so that with your collaboration we may yet do it. """YU 6 1~" ch^^c^~ THE UNIVERSITY OF BRITISH COLUMBIA 2075 WESBROOK PLACE VANCOUVER, B.C., CANADA V6T 1W5 DEPARTMENT OF MATHEMATICS SoA fAA^ Aou^t y LeO Ot A =0 cd, L4L caAjt 7 Je ( #L p3 ,(t7 Su po se X u;OLI"AA4A ^;f /19 aAI^t r o = 0 L S(V s>,o) E 41 n~ *k% , CA)V 4e'. aO4 LAtA v 7 7 V c&n a {An it J Tenaa ,C,. Lal kr, ~k ;L~ ~) A"q = S _ p i. o cs "J U(A/L 1= K= X  5f_ E S~t ALi tIaH wanA . )n;n4OA lU ad BexutJ " IEAAIV qj 4Ii  l +to p acJ 1JlA Sk. 1oi dLAV g, f "T oC ~~2.3 doK,% . ( FRA (jj+,J L% 6Pu t cause, a a'll/^ S S hA^ 0i L~.C)~.YcL& 1 (4 ~L(L L~~3 Lzt&hLf bY4 A/JeAJ  eof uJIS L ^yfvc^ bUA sc I, 1 )f L e+h~ iA . o a UA"<4UOL c7 t9 .. j{ X ^ o. ^e j)WP = j no ;n \J . /^J T'los AAd ~aw A Y, ) , 11/13/98 FRI 13:54 FAX 604 822 6074 .e..... ___r Lk . ...... ........, .. .. .a... t .........o b... .. or , .... ..".. .... .. . o Sa e A I.... t9, ........ ..N4; /o e.. .. sK c I6.a .......... T L_ .o .._ _ay_ L.... S_ __ (. ..TAi4 .Q ._. s r e .l. .. .._.. ........ ~.. a ....._.nv s............. .. ._._^ ^ __ K. ... L o ov ...A h .. ... .. b, ..... ,, o.._ _ Un s o t ......or .... s_ .. ... .... _. .....  .. ..  ........ i ^ T .K  _...o ...... .. .  ........o ..^ . ." ...... .... ..._. Ss..7. IY, X . .. T $ ..... . (' : >, j s o >^ ,e> ^ l ^.AnsA ...... ..... ... ... ............ ... UBC MATH o002 11/13/98 FRI 13:54 FAX 604 822 6074 UBC MATH I2003 ___ s 3 n a D.. Te ? .... L ........ ___ ..__........... ..... .... ..6'.. ....).. ......... .._ '.  .. ... _ __ t ... 1 i ^..l /.... .... ao(A. A.... ____ __ ....: . _____. ._ . ........... .o... 0__ O._ ... .. f . V St )....... o ;. .o ._____ .J.. _. __ ,.... _...._. 4 K ___ ___Le). __._ __M Ld 1 M Wmmnia@hotmail,com From : "John Klmmel" To: Subject: PublIcat ti Date: Fri, 7 Ftb. 2003 08:37:44 0800 Deor Farid: \) P~f8 i (AtJfiaJ "^ ^^StS F~L~# saL/0a~f Inbox I Previous Page f e t j4'I 'ri AO{f ( 7f7 fY 1! 2t 2 ^US^OIjP ~r ^^%^ i /f2aae {~e~ j. P ~Z 4~ 11e ~JO~rP~a4~7 ~16 ~IJ8" a 4 B A/ 7~~ QKr^ f/Ok ^u; ,. c;P Tf4Le ~it /2~C IQ~~2~U~4 ~41 / t~ 7 rd4.. $ B6 p.01 0 *'*" ?Xe (t4 a, ~i)~l3pQaL"^ ~% ?~ ~"B~C: rc~id~~e~s, "4l k ~c.: A Lo qlvd <4er Jo< fr; < (1A /AzW^L iAOc11 ^^~c n^Si^uh^ iS'^'P ~" 0 (%2 Prof. John Walsh A Dear John: Here is my initial reaction after glancing thru the first 6 pages. Faites cannot be rendered into facts, so better just use Englsih such as Generalities (same spelling in French exceot for an accent aigu). Now you say thstsection is all about separability. Look at the Index of my book and see what I said tb@4V For ex. I am not sure I men ioed "standard modification". If so how will the poor reader (remember$ a student not a professor of sorts) understand what you mean there? How clev er that agini function is separable' But that's what I found out, noyour acher Read my article "ProbH ability and Doob"7for more details including a note on Wolfowitz, whose son Paul may wa get you and me killed by terrorists. No jokei since separability was not in my Lectures (except ip homework, see Index)o If you need it you better explain it. Next, "maasuarble process". See Index for Borel measurable: Sf they are the same, why not refer ?/You can use "stopping" and need not make high tech. wBrk too hard, why not just say I used another adjective originallyy your teacher). By the way did you define your stopping sans tribt? Better be precise. I had my first chuckle when I read you made such abig deal about Hunt's Lemma. Read Th. 9.4.8 in my Course, supposed to be prerequisite to my Lectures. Eaysprof". Is adaptedj ever defined  not used by me. "Resolvent" is mentioned but not used by me. Not enough to say your U is yxx Ua IN THE BOOK elsewhere. Since you Ray this and that, you must tell the poor reader tn t why resolvents in lieu of my previous semi group,"must be" used? I remember I said something therevaais ;t,, The tex(merde)may help you later but it will save the environ ment if you pay some attention to other parts of the same] book when you must use Frenchy words. PaulAndre is dead. A final and serious cpnseii du style : on p. 3, Iseparable: "by looking at ...". On fa t tell choses debout vant le1 ort (?) accompagn4 by handwaving, but it does not mean a thing.' 9za I prefer fax to phone, for the record etc. Owing to telemar keting, phone callsW are usually ignored: attention. / d ^ I dE f d / \/cxa I 2/18/2003 Prof. John B. Walsh [There is a John WalshLhow ,I have mot Section 2. Def of supermedian, 1, 2, ... ? J Ke . Did you ever tell the student (') why Ray and not Hunt process as in the first part of the SAME BOOK? And why Branching prhnts? Where do they evr bugff? How come Gil did not want them and what have they to do with electricity or finance? A Preditable time  remind me what 4ourf jpint paper about Meyer's theorem is all about? it Ci I Z f.i p. 14 and before: Feller prcerty)f (see my Index) 15: SS I don't accept this notation 16: StonpW The rem (you have the Theorem sometimes): I do not expect MY students to know what it is. Refer to some book in the Bibil. Do you really need the fancy result? Wouldn't di. alone do A? p. 17 and eaier: before any talk exhinit a nonstupid example gm of a branching point which really shows up. Not the kind of/ Ckw junk Blumenthal used to cite. That does not work. I invented Moderate MP, ~iA t bugged me when a Markov CHAIN path 'I i atsn when a Sy hits a "boundary". w ad this virgin remember it. ,lt p. 20, I would delete "simply" an EXPLAIN the next sentence Where is "inaccessible" defined?? Who knows what it means. ahs eal Prop. 2.3.: isn't this just MY DEF of MMP? S Polar  refer to old text'. Q Mm Sec. 2.4: must say martingalee on the SAME triple "otherwise it doesn't make any sense./ p. 22, Lemma: k is misprimt p. 25: You expect the student to know Prohorov's th. I used to Know but never, never needed it. h0llc JlIo I (a 6V p. 24 Refer'Upcrossihg lemma to MY COURSE unless it is in Sthe booked) p. 26 (Tt) needs brackets r T A/u4 /0 4 SNow I wonder if all that jazz applies to L6vy proqc If not S what's al f&ss for, Hunt wrote his memoirs to APPLY to Potentials. He failed in his second memoir because no reverse/dual which led t~ BluGet's'duality hyp. Now if you have "Time Symmetry" can you CIO do better? otherwise why all thertpreliminaries? Ca fCPJ a/ cO*3 . p. 28 How about"a few"6rds're excessve/supermedian/.C.F. Prop. 2.8.2 Isn't this just CartanDoob ME (I have a short version in Meyer's SeWinaire  check it?)' 0 r'9LR a 4 c^ ^u ^1/pL t^^Lf^L^I Cl 1 1\ i/_>; ,. TRANSMISSION VERIFICATION REPORT TIME : 02/10/2003 07:10 t foY J n A YAl? l EL UR p. 30 What's Ray's flawr sa i Dh[E,TIMe 02/10 07:10 FA N /NAME 1 108341 065430397 DUR ON 0: P 01 SULT OK MODE e I 'o tP ST DA Here are random notes on scanning your MS. I sent yojsome already P. 30: explain Ray's flaw?  P. 31, I never mention Lusin : what is it? Why do we need it? Did yoi know Luz as accused of counterrevolutionary by Kolmogoroff et al. and was barely saved his life by STalin. Where is the BEEF? That's the main question. I have forgotten all about Ray etc. and my text is all about Hunt process and more spe cial ones never, never even the silly "stand ard" one. Thus for the student of the new book there isa quantum leap which the book must be ware and prepare him for. I see few signs af this. You are just ad dressing a new crowd. For those like US (many) who want to learn pro bability for possible applications to potential (Doob, Hunt, even the dead Meyer) the first t irty/forty pages look like just preparing the dough  so, once more: WHERE ISTHE BEEF? You have written it and of couse can keep it*,but you certainly owe the beefeaters a ot, repaeat:a lot, of explanations to help them see the "relationship" (these days on TV shows it means SEX  the realthing') between j1(y Co acht4;'2 * *fcyWc/V2 all these new ro1olin OVa und foreplay to be recommended by Phil, S that taEdNo. 1 show on Channel 7 here, surely also on yours) and the EAL THING that Doob, Hunt, et el engaged in and exposed in my original text. Take a peep now: there is fun there too. p. 38, Prop. 2.11: what is E, and F? p. 37,Is "useless" a math. BEF. If so where is it defined? p. 48, Th. 6, ba6 most of your props have three numerals: is it not confusing to havesay31.2.3, then 67 p. 53: "time homog." Statt. trans. prob. Choose ONE. explainn 1 esy' double reversal  may be easy for John W., not for oor student. EX. in (3.9) you have het and tilde('a? why both in same formula. I have forgotten what's what. w; pj hen I get time I will try the crunch on time reversal where I seem t recognize my old symbolism /, I don't promise I can read it altho' I recall even that jerk David Williams in his book "praised" MY style of (position. Let's not spoil it via Ray etc. 6s!A ) URGENT Prof. John B. Walsh, Math, UBC End of February, 2003 SDid you get my comments, two pages sent twice separal ? More random notes while scanning (only) your script: p. 30what is Ray's flaw (he was my teaching assistant) p. 31: Define Lusin space and EXPLAIN why we need this generality. Even Bourbaki claimed to be "moderately general. y MAIN QUESTION for the first 30/40 pages: WHERE IS THE BEEF? My book is all about Hunt process and more special ones. You can generalize as much as you see fit, but you must prepare the student (not professional) for the quantum jump from mine to yours. I see few signs of this. Too lazy to look at my book and A too messy to explain the raisonetre of your preliminaries (fait es diverse et pourquoi9J Your teacher Doob, my friend Hunt, even the dead Meyer, did the processes to apply (MATHEMATICALLY and ahysjcally) to potentials. As I told you last time, Gil had to restrict, specialize his hypotheses in his second grand memoir to do some of this, later further restricted by miser able duality hypotheses in BluGet. Maybe you have done better by yout time symmetry? If so tell the student where all this fooling around with spaces and compactifications and resolvents come in? Sounds like a lot of foreplay (sure, recommended by v~. talk show hostess gprah and host Phil: have you watch them late ly? I did not.) without the REAL THING. Do it alright, but try, repeat: try, to motivate it/them a lot,lot more and tell the frustrated unfulfilled audience why and how it/they help the RELATIONSHIP [nowadays I can't turn on the radio/TV for one minute without hearing this word three times plus another three "INCREDIBLE" ]. Take a longer peep into my book and see how your foreplay can fit in, or else skipped? You owe the student at least a few more references: use my Index. p.38, where is E, where'is F defined in Prop. 2.11? p. 37: is "useless" a math term? p.48, how come Th. 6 does not have 3 numerals as other propositions? I can't find Th. 5,4,3, ... P.53 "Time homog." is "Stat. trans. prob." SAY it openly. If that double reversal is so easy* SHOW it. This reader is totally confused with the double use of hat and twiddle, e/ g/ why 1has ', etc. etc. I'LL stop here and wait for a return fax: do not phone. se e. e. (3, C 398 I SUPPLEMENT: MEASURE AND INTEGRAL (iv) Let An E 9, An C An+i for all n and A = UnAn. Then we have (35) limE(A,; f) = E(A; f). n PROOF. Denote f by (33), so that 1Af = j bjlABj. By (i), E(A; f)= bjy(AB,) j with a similar equation where A is replaced by An. Since A.(AnBj) Lt(ABj) as n f oo, and 7=1 t E l as m f oo, (35) follows by the double limit theorem. Consider now an increasing sequence (fn} of basic functions, namely, fn < fn+i for all n. Then f = limn t fn exists and'f e S, but of course f need not be basic; and its integral has yet to be defined. By property (ii), the numerical sequence E(fn) is increasing and so limn 4 E(fn) exists, possibly equal to +oo. It is tempting to define E(f) to be that limit, but we need the following result to legitimize the idea. Theorem 8. Let { fn and {gn be two increasing sequences of basic func tions such that (36) lim 1 fn = lim f gn n n (everywhere in 02). Then we have (37) lim t E(f n)= lim E(gn). n n PROOF. Denote the common limit function in (36) byf and put A = (C e 02: f(w) > 0}, then A E . Since 0 < gn < f, we have 1Acgn = 0 identically; hence by prop erty (iii): (38) E(gn) = E(A; gn) + E(AC; gn) = E(A;gn ). Fix an n and put for each k E N: Ak = o S2: :fk () > gn() . Since fk I fk+1, we have Ak C Ak+1 for all k. We are going to prove that (39) UAk=A. k=l (60ou; (o07z URGENT: Prof. John D. Walsh, Math UBC Re iVMarch 4, 2003 Dear John Haven't heard from you (fax, don't phone.) since I sent you successively tree fax(es) of my reading notes. The last for identification purpose)contains the KEY: FOREPLAY. Now I FINISHED the whole bunch. My comments are meant to S43estL only to be suggestive, i.e., many more to be otedin like A ( osi'fer&! mode by the author yourself. It is not exhaustive_ _ p. 60, line 4: "since ..." I do not see. .%/ (t) p. 61, 1. 8: better "...for which t it is rational ..." A p. 62, line 1: I am utterly lost in all the U with and without hat, twiddle ... By the way is there one without any of these? "IjS ces nofef,', i' p. 63, 1. 8: I assume you defined before mupolar? ire] Hunt's Lemma  did you not make a big fuss about it and it is in my "elementary" text, I to/ y'tl. p. 65, Prop. 3.2.1: "compatible" defined where? Th. 3.0.1  such a BIG THEBREM with THREE numbers? SO how come you had also some (not so big) Th. with just one number, as I noticed and warned you about in an early fax ? What is your numbering system? Even Bourbaki worried about theirs. p. 67: I can't distinguish the two Tis there. CaQ) yogr2 j0 I seem to recognize a bit something in the reversing theorem, at least that small "el" j used. Do you know I did it with the same "el" in my MC bbok, where reversing from the first infinity was carried out? FOR THE FIRST TIME IN MATH (cf. my sly historical comment on dead Meyer's earlier effort in the book that earned me a good piece of canadian bacont. But now you cover up everything with so many hats, tild4s, topdashes, ... I could barely see the forest. Let us all hope in the nex Chapter some real screwing can be viewed? Maybe even a little advancement over the ew re sults in my Lectures where reversal/duality is visibly CwJ ) exposed?_ Some such thing poor Hunt his pupils, the japs, ... tried#to follow up? By the way, how about asking A Dellacherietread parts of YOUR book, he is alive isn't he? ,Q u Your other pal Well is "useless" (ake your undefined set), 4U1BC1 but maybe that other probabilist(!iCaw Ym6ght ) can help? Add has / y I 4f' can't owing to age. Knight would like to see his IqooK name there but I wonder if he will really read it? W k LC / S/ A;mn 3 MEASURES IN R 1 393 If we intersect both sides of (26) with the complement of (ak, bk), we obtain [b', b] C U(aj, b'). j=1 ik Here the number of intervals on the right side is I 1; hence by the induction hypothesis we have F(b) F(bk) < (F(b') F(aj)). j=I jAk Adding this to (28) we obtain (27), and the induction is complete. It follows from (27) and (25) that F(b) F(a) < (F(b) F(aj)) + e. j=1 Beware that the I above depends on E. However, if we change 1 to oo (back to infinity!) then the infinite series of course does not depend on E. Therefore we can let E 0 to obtain (21) when the "=" there is changed to <", namely the other half of Borel's lemma, for finite a and b. It remains to treat the case a = oo and/or b = +oo. Let  00 (oo, b] C U(aj, bj]. j=1 Then for any a in (oo, b), (21) holds with "=" replaced by "<". Letting a + oo we obtain the desired result. The case b = +oo is similar. Q.E.D. In the following, all I with subscripts denote intervals of the shape (a, b]; 7 denotes union of disjoint sets. Let B E So; Bj E So, j E N. Thus n ni B= li; Bj = Ijk. i=1 k=1 Suppose B=00Bj j=l so that n oo nj (29) EIi = Ijk i=1 j=1 k=l (0 0) f bl0Fq Prof. John B. Walshf March 7, 2003: 7:15 AM Let me hope your )ax is not BUSYjNO RESPONSE at this hour, 0 'n tod6ow I am sure you wont get b=t5 ti  RELATIONSHP. RELATIONSHIP, RELATIONSHIP. /^ ) Call my old text A amd your forthcoming B. Not in Preface, NO, but in the Introduction you must tell the student why you needt spend 50 some pages on all that jazz: lesolverts Compactication, Branching,..., EVEN REVERSING [TIME SYMMETRY in the title!] to DO or OVERDO what is already done in Part A, without any of that junk. After all, Hunt did not need Ray etc. Exhit as many as you can easytSsee THEOREMS covering more ground than those in Part A/ And TELL IT not only HOW but also WHY Part B is worthy sequence (as printed) to Part A. The buyer should wonder. By the way, take a peep into my INDEX to see what Part B will RELATE to Part A. I saw those symbols ( A .~ ARE they going to be expanded in Part B? Remember "nearly Borel" A have you Ray it away, and what about Section/Projection/Capa city? As I told you I can't read it anymore so .the ball is in your Park()' Out fax worked perfectly" (650)857020.on Sabbath too. s At C  3ik J^u^^. 0 Saturday Marc# 8, 2003 i Office closed, No Fax SHey here is where your BRACHING should be illustrated. Take a A q look at MY MC book, 2nd ed. where I not only reversed the path From T/ but introduced that fatal el that will now appear in another 2nd ed. Now if you have a copy of my Acta Math.. xx 1962 [did you exist then?], MMP was discovered at the last stage of pageproofing. 40t/ years is a long time but I seem to remember my futile struggle to get SMP at that "Branching" and nearly faulted. Grace a la den're epreuVe* it suddenly V A /C ap proysMNGes A Nearly ipno this great Italian bon mot 1. Your blow by blow notes on my three earlier notes cheered me enough to plan a "SANS BLAGUE" (pre)historical epilogue to the section in Part B (perhaps also a hint in Part A) where those miser in PartB B a able branching took so much space and time. You should EX A^ A A<'. i PLAIN how that first "explosion" (FellerDoob) and the tri E vial special case of ghostlike return to earth (my words, Your teacher's "stunning" performance), as well as the COM IPLETE picture described in my Boundary book long sold ouI gL Andre cooper F Par example: when the path cpmes down frm ao (a boundary point) either stickly or nonstickily (fee my Bpundary book), is IT still a branch* ing Apointi? I do not know but you should spend six Itx seven lines TELLING the reader what's up. This is what I call a REAL example not those "uniform motion"9ye A junk. I am typing thisas well for my own MEMO when I ;x(I my Part A referring to Part B to be filled in by yo ^ A et& 4AC p n vu Sunday March 9? ,2003 Dear John: I am inclosing two old reprints you may not have. I assume y&u have reprints of our paper on reversing. Iust read the first few para graphs (recall David Williams in HIS BOOK complimented on its clarity] If your Part B will contain most if not all the stuff there (too many pages), at least you shoudl give some "APPLICATIONS"  of course Mathe matical ones if not physical. Otherwise why bother? tay think so. Re my old reprint on Hunt's Hypotheste, this seems relevant to th4 first fourtyfifty pag es of Part Bt much improved of course? But let me know if the results in Part B have anything to do with wyat I did there. I gave the talk in Rome and on my return stopped overnight at Ovieto, danK the marvelous Orvieto AMABILEarecommended by the vineyard owner who is(L _t nor on v' USA & ca nada) also the owner of the rand hotel I stayed in and who came to my table to greet me. Those were good days. My ntop was at Strasbourg with the Walshes  did we eat at *Zimmer', (maybe not with you). .r eve. Cro c.< letx I did not thlAk I would look at those old papers again, most of which I do not understand now. Caest la GUERRE. I have only begun to read your response to my "notes", do not Expect to understand all of it and will not make further suggestions. Tdg(t follow those Frenchy and probably did not ever use words like adaptedd". If it is not in the Index then you must define it, ditto for other terms. Meyer is dead anyway By the way/ are you citing their encyclopaedia anywhere and is Dellacherie still in Rouen? I alwefs want ed to see Rouen* By the way, are you going to France this summer? Those 4udes will probably start the war when you get this letter; some patriots are already boycotting French wines, but you are Irish/Canadian. Tell yoty resigned minister that "moron" seems appropriate. Ciao. L s)ir i'4erc? C1/AY /Ae( a5k 6h "* /0 0PrrT. 3/12/2003 Prof. John B. Walsh I sent you a batch via IsoJLe which will take longer time. I've begun marking stdthe numerous minor corrections in my old copy, and found out things you may want to refer to. On p. 136, notes on Sec. 3.8, Ray and standard are mentioned, so now may have to refer forward to where you define those, if I remember Y More import ant for the reader is for you in your Part II (I changed A and B to I and II) to tell him where you go from Ray etc. backward to my Hunt. If and when you do serious theorems in a more general context than I did, it is your duty to tell the reader .... A It may help the reader to print in large letters a Notice like: From here on we will deal with a MP with the following properti es: (A) (B) (C) (C).... Cf. my list of properties for BM bt> fore its special results. e I,, . Both "Adaptedtkand PStoneWeierstrass" are in my Course, see Index. A special form of the latter is spelled out and sufftces there. By the way if yourUBC is not bankrupt you should tell it to BUY a copy or twoTthe njENLARGED edition of my Curse with complete exposition /f me'er Rz ,oi) exposition f mear theory. I have a couple of ccopies for sale (with discount.) to anybody who can afford it. Fin4 HYpothese (L) and (B) in Part X (formerly Part A) and let's see if you can get away without them? Does not your junior college Ed Perkins(ii& Barlow) have the capacity to really read the new math in your Part 3 and possibly find errors and make improvements? I can't. Fax any reply, No phone. USA(650)8570208 L'LC ;OLF naO71b URGENT: John B. Walsh March 25, 2003 This is the second fax sent you today, while you are sleeping. The first is to ask you to rediscover Boob's no Projection proof of Meyer's virginal effort. This second ona is a slightly less old nut, Theorem 1 in my Stopped Feynman Kac functional, Seminaire XIV (1978) Strasbourg, re printed in the Second Edition of my Green, Brown, and Probability,t/'Z p. 166 Footnote. Sine our new book has "time Symmetry" in the tttle, we should give a new proof of this obviously time symmetric result. I've prosed this venture at several inter national gatherings, the latest maybe at Cornell where that Russion guy who wrote papers a yours (name forgotten) was A and I challenged him X*Bthat. Zero answer. Hope with all your a paratus in Part II you can do it. It would be a true bench4 mark". It is true and proved (by me in another paper) for all onedim Markov process with continuous paths, sometime misname ed "diffusion". [By the way if you use that term you better define it.] Once it is probed by reversing the TIME, I am sure you can extend it to more general processes. By the way the function q there can easily be generalized to integrab rable etc (I assumed it to be bounded). Let's see if all that jazz can Ke used to kill a fly? Bonne Chance! Why not try it on some youngsters like your colleagues at UBC? 4 I have copies for sale (at a disocunt) to your libra ry or other customers. Is the First Ed. in your library? The 2nd has a lot more stuff. ( Dq) URGENT Prof. John B. Walsh 0? Hey please sketch (not too briefly) that famous proof on that flight from Zurich to Beijing and send it to me for possible rewriting. I want to add it to that anecdote in the Notes so that space is avail able without pain. Your teacher published it somewhere but I am sure RD A you can recAver it. That will be great fun in the also Hawkes's remark about duality.,,CrIreversal. new ealtion. au Can you show some nri* ^ r Tim now with your big buildup?? Bonne chance. Reply by ^ o by^i (Atc saUxMH f 1. iP!P I n aunoald qornui n AuAq no0 doq I 11 psai pui doo v rnq jaoiq pgq noA tufl.ag o0 qP UZ wozj oualdn oqi au poIaddeq iqtW PuqN mouno, 0p1 SnoqS AMOU 01 lUM noC j" mnoqtnonq pu9uuos lTuzql q sium poo a pmdnwl pru ulopaouam m u3.t.!iM unoouwnu oln saq tlunqD Jossajojd Aq 31ooq moJ paodxa pinom oao sy qoitopw suzsqojd oa uoponpoJu! uu qii1m lofqn i 9ujojt o o I)op! ia bmuqmoo pooq .qj**uiqojd p9Alosun omos jo @pIU o05 119 uoumO pug 'palmnuo aq o03 p1q OAtq qMopA Kdo p*aUtApi 0om0 q)im dqsqonlpj Jo uorinnoup jtUnzl 1 sumrluoo uott3M qowg "A.qInb Iq .nq oqi jo uoopuodx .ql si uow1 ip iV 'tooojd Anp s jo o03u 3 joPdmn m oqu 8mo ipoA pua '(pui Sunlmnb o1 paoueqq ameu m! p4q snq au!i i!xa ISej Iq1) Wn. I!xo Ivt 0qI jo 9191 Oqi lumirqduwo 'apwop l91 aq04 JOAO ~(JOm uMo .Stunq3 uo p~mmq s! ojaq Iunonawl aqI "jooqiu pn.uqiod jo uounon.p oi0 polo:Ap si (,siuomdo(pAaQ ltflulood.) 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NO iS3afil3o31 s^Z ^ y 68L88 1 (4196) 9L A;eooog TV~OTU1qHzW (U.fTUUT1.qigo=J) uopugo e*q jo uT OTTnI 3/24/2003 URGENT: John B. Walsh March 26, 2003 Five pages about M. C. are here, in fact some duplicates came earlier. This revue is an excellent idea. The only "probleme" [the English version of this good French word is being so "incredibly" (another fantastically abused word in the current American speech) abused that I hesi te to EMPLOY it, but the French word means what I really mean.  OK the only problem is who among the readers can appreciate your Examples without a rather deep acquaintance we MY (hey) two books on the subject. Let me suggest a few particular REFERENCES to help those few readers, later. For the moment, I as sume you are working on the two Math Problems [correct usgge here] in my last two faxes, one in each. The second one as an Ex. of re versing may be too hard even for you. [That Russian said it was "unfair"! What's his name who was at Cornell then, c. 1994.] So let me just mention the first about poor Andre's theorem. I trie ed to read it just now but had to quit. But please tell me: (1) That fancy projection around p. 40 of Part I, using analytic set and all that (forget about Doob's airborne dismissal of all that stuff): are you using such advance ced weapons in Part II? If so yoL better say so. (2) Assuming you learned all that stuff in Strasbourg, can you tell me why Meyer told me c. 1983 in a letter: (i) progressive measurability is unimportant. (ii) that theorem should be called "section" rather than "projection"? If you can tell me I may change the name there (in Part I) though it is too late for Andrd. How many times is it used in PartII? If you do not do analytic sets are you just going to do as I did by handwaving? As you see serious questions of prerequisites for the new ed. may be involved here, since Part II seems on a higher level of generalization/abstraction than Part I intends C'est la gue rre. rtotbe) VL Ckc 6 y < 2Z 0';7 Not.too urgent: Prof. Walsh M March 27, 2003 Dear John: Re M. C. you should say a few words abotu the differentiabi lity ,f Pij(t), about qi= oo, the case of "conservative Q", all of which may be referred to my book and some may be more ready in my Boundary book. Meyer did a great injustice (somehow I did not catch it then) in calling the diff. th. Ornsteinl, It AustinOrnstein. 0. proved the hader general case which however is (till now) useless. It A is Austin who proved first the useful (for your Part II') case but my later integral representation may be more relevant for aum purposes. Anyway unless your part has a lot of diff. eqs (I have not seen any) that thapter'of differential equations should be motivated. Bru sent me the big MS by Doblin (the way he wanted his name spelled) now al ready prinj (see his mention of Austin I t4ld him: Yincent D. never did prove that! Curious. Nor Kol.eV al. It was On my AFOSR pro ject'r0 t I hired Don and he surprised me by proving these thinSC. (Doob did not, could not). Take a look at my MC book and add a few historical words in the Notes [there's where I amuse$ myself. I will state my reversible theorem for FK functional posed to you: that Russ was a student of Dynkin's who hid behind his master at Cornell but I did t6Q1 him the problem. Ydu do not know his work on reversing etc? A long paper he sent me but I forget even his name: ask someone else). About your lost MS it was the memeo S4 copy of your talks in Paris I think. It should be relevant to II, 4 i6 how about your published article having to do with last exit  I forget the theme but expect it has to do with II. Anyway we i e0 should finish by June whatever you add. Y subtract. As I realized l I can only make minor corrections in Part I, add a few cool remarks nhy ? [Eggd ftr.Hawkes to say so, do you know him?] and LEAVE THE REST to Part II and you. You can also say whatever you d)n't ROV( in the Notes for your chapters. Are you having the math checked by someone in UBC, and how about Salisbury? (I ; ~ (C URGENT: Prof. John Walsh I sent you yesterday my comments on ydur Examples with Chains. % Here are a few more page by page marginal notes/ J htst s&kp MOL, 2, before Kol equs say that EVEN the qij's requii* good work by Kol. of course Ref. to my book. Now tell why we need those stupid PDE? p. 3: Ex. 1i when the path reaches co, I WOULD remind the student that do is NOT a lumber hence out of'space. RefeF then to your teacher's ghostlike return from there yonder: of course in my book. A whole section is devoted to the minimal chain there. Unfortunately when boundary points etc. are concerned, Ref. to my other book is required. A few specific page or section references should be given. yralLca' / e e" If all qi=qii are finite, we have right continuity (proved in the 2nd book actually first noted by Pittenger). I forget what we can say abour the rest of Hunt hyp.? Maybe you should say p.3 p. 4: Are you going to tell how boundary POINTS can be defined ( o Ray et al.) How about nonsticky bdy pt and explain why a sticky bdy pt does not branch? Hope y6u will spare the pain of Martin bdy?? Even that is mentioned in my first book, I think. 5 fte sllOdj bra~nkC p. 4,lower paragraph: I don't know if Yt = f(Xt) is Hunt when Xt is: surely we need some condition f? This is the.kind of "change" (not A TIMEchange) not covered in general theory? By the way so far I see no Ray compactificatopn. How about an easy example? p. 5: suddenly you'say "stable",not defined. Unstable is called instantaneous in my book. What do you do when all sltes are inst.?? How to Ray it if you'can? That Ex. of a chain living on Q is Levy's  look up a whole section in my book Folks like Henry says. as you repeated, it is like a diffusion. I do not know why and what is "regular" diffusion? It would help *f you can show how Ray compactifying helps in those examples/ of Ca .tiS, bttf ey f~v I call "well meas." "optionally meas. see pp. 4344 for this and A the Doob incidents If you can't recover his proof please ask him for a published ref. "predictable" is adopted in my book. So "predictably means " Do you need strong Feller property? I would like to wean4 hvea a% icSi4.it.j ,.Pe'aatmy article "Doubly Feller .." SrryI i in the Seminar on St. Proc. zAfter killing cS 4 es. ou saa domain a Doubly Feller proc re ~J pit S mains so  n standing p\eblem ... ._ .EGt? Can you employ it somehow.. 0ss1 rc Lk 3N hit, March 28, 2003 URGENT: Prof. John Walsh March 28, 2003 I sent you yesterday my cOmments on ydur Examples with chains. fX Here are a few more pagebypage marginal notes/ 1 hkat sop nu . p. 2, before Kol equs say that EVEN the qij's requiz good work by Kol. \of course Ref. to my book. Now tell why we need those stupid PDE? p. 3: Ex. 1. when the path reaches oD. I WOULD remind the student that cb is NOT a Iumber hence out of'space. Refet then to your teacher's ghostlike return from there yonder: of course in my oo k A whole section is devoted to the minimal chain there. Unfortunately en bouna points etc. are concerned. Ref. to my other book is required. A few specific page or section references should be given. Ef sya% reJl c't IPC4,asC e . If all qi=qii are finite, we have right continuity (proved in the 2nd book actually first noted by Pittenger). I forget what we can say abour the rest of Hunt hyp.? Maybe you should say .3 p. 4: Are you going to tell how boundary POINTS can be defined ( Ray et al.) How about nAnsticky bdy pt and explain why a sticky bdy pt does not branch? Hope you will spare the pain of Martin bdy?? Even that ts mentioned in my first book,(I think. I ft aJoyl b&raf f' ~ Pa. cJ p. 4,lower paragraph: I don't know if Yt f(Xt) is Hunt when Xt is: surely we need some conditio f? This is the.kind of "change" (not TIMEchange) not covered in general theory? By the way so far I see no Ra; compactificatopn. H w about an easy example? p. 5' suddenly you say "stable",not defined. Unstable is called instantaneous in my book. What do you do when all sates are inst.?? How to Ray it if you can? That Ex. of a chain living on Q is Levy's  look i a whole section in my book Folks like Henry says, as you repeated, it it like a diffusion. I do not know why and what is "regular" diffusion? It would help *f you can show how Ray compactifying helps in those examples/ &f ChainS. a er +rn otP I call "well meas." "optionally means. see pp. 4344 for this and the Doob incidents If you can't recover his proof please ask him for a published ref. "Dredictabl." in adnon iAn 4m" "bk  "  .is adopted in my book; so LPred ableameaske^a ]l ub P e ler Process" in Sem. on St Proc. Please take a look at my "Doubly Fee process in Sem. on St Proc. 1985. A Hunt proc can be constructed from a Feller semigroup. Do you do such things? It is proved there that a doubly Feller proc. killed outside a domain remains so. Can you employ this somewhere? 1ead the little history discreetly mentioned there. It would make me feel good if it serves in your Parn II (forget about the functional). ( & ) 2 7/ Prof. John B. Walsh P/elt e e(l er fr d y~y March 29, 2003 In the good old days departmental offices are open in the morning of Sadday. Do not expect this for Vancouver but this is URGENT: Part II MUST refer to Part I fir the material (faites) used! Sometime ago I wondered about StoneW. I was so forgetful. Of course I used it plenty, e. g. in Sec. 2.2. [Many trivial misprints like the E in the wrong font.] Of course I proved in meticulous details that Sa Feller process is a Hunt process. I almost never "construct" but SWa there I had to use separability and apparently resolvents. You should pr ov : prve really take advantage of this detour to implant the Ray doings etc. q/1 This is what I called RELATIONSHIP. It would seem that you have not troubled to sc the contents of Part I to make those RELATIONS. If that be true the new book would be separated union. Pac b nV I mentioned Art because I still remember a little surprise when A he told me. In my book, a long derivation was wri en out. in Frenchy _UyT4_. style by the dead Meyer: if all states are stable, then the paths are a. s. right continuous and have left limitff. Bravo' Of course I did it in my Strasbourg class and it probably gave Andrd a shock too who was used to swipe in Luzin... spaces. 7 of courseoo is added. 2s ,t 14&t&L ? This is actually easy when we know that each stable state is taken in [C,Os and there are only a finite"of theft in the finite Levy's great "obvious" observation (he could not prove it; I did) that Doob totally missed in his efforts around 1940. I should really add a cool historic note to this (I have probably done so elsewhere). A suberb case of intuitiontvisavis "nonintuition" (for lack of a better word, not e. g. "reason" or ,worse "logic"). Of course I did NGT bother to look at 1 your handwaving in your fax about this result. Tell me/ 9 wYn Regarding my concern with PDE, it depends whether they play ?,A any open rAle in Part II. Only Laplace's ~howsup in my part. Did I OLot ~d) say in one of my Notesthat Andre (N., not Paul) probably as a young assistant felt obliged to show his Academy bosses (Petrowski came to my mind~' read his easy book, but there are many others in Moscow) that "the stuff" he did (c. 1930) had something to do with th fkain Stream. So he wrote down those two PDE but then of course saw that he could noway solve them, even INcompletely. Seed his "Ueber Analy tishen Methoden" and p. 26 of my Random Time book sold you for a poubd of bacon. l , S IA~~ Tk o t) I (sLJ) c qI Td H 6 ?4 { l3s E Co, t] S4" j cm aS (s, ) C3H r. O4 c L4zub Q FiI SSe CI o I 4 \ KTZl( 1'f t >% It si~  1 a 4ty Ilw cCI~ (i ,3 o) URGENT: Prof. John B. Walsh On p. 2 of my LectUres, middle of page, I dropped the word "adapted". I forgot. Now that you give examples with chains, it may be worth telling your reader that even though that process may not be a Hunt process in fact I allow instantaneous states (maybe all are so), a STRONG MARK PROPERTY holds in the obviously best possible way. Namely whenever the optional place X(T) is in the original countable discrete space (no compactification). In fact a more specific form of i is proved in my MC chain book which you may want to take a look. I need that specific form involving the optional timt T for many applicationE Unfortunately I have forgotten where and why. I&csw In the ae"R"6le 2i 2 more case you heea deatJgg with/such things Sa needed. Do you have a similar situation where the SMP holds where it makes sense I mean if 6 is added to compactify the space, I do not define P(6,.) and of course the usual form makes no sense. An Index for ypur part will be necessary)including all the symbols ypu use (which may not mean the same as mine). By the way, I still remember Meyer's saying that progressive measurability is not important, the more refined optionally/predic tably are important. But in my exposition we begin with a prog. meas. proc. in order to refine it to be the others. Do you do the same? Maybe I did not understand what he meant? Do you? 4/7/2003 URGENT: Prof. John B. Walsh I forgpt I dropped the name "adapted" early on on p. 2, middle of page. You should compile an Index for your part including all the symbols used. If they mean different things from my part, better say so. )To It is a good idea to include those examples from Markov chains. After all this is the Horse's Head for us all oldsters. We stuA with Sir Issac conic sections, polyhedrons before the followers make up Luzin et al things. Maybe your latest stuff can also show a new way of doing the Strong *rkov Property for a general chain? In 1956 I proveahis property obtains (fancy English') at any optional T whenandwhere X(T) is in the original countable * space without co. Quite an achievement that the Russians aX~i* Yushkewitz did not do )Dynkin (they left large holes in their effort whiCh is also published). That was just the time %ta Ray was doing his stuff  he was at Cornell when I gave my talk on thl' stbject(SN/p) and took me home to have dinnerI think afterward. So let us (me) see how you get this particular case from your new presentation. Don't worry: just show as much as you can. But here is a question that had bothered me in the pastwhich you may answer (for me if not for the reader)" Take a look at my MC book, 2nd ed, p. 179 [if your library has only 1st ed. that is no reason not to find it there on another page] I went out of my way to say that, around line 15, the theorem with several corollaries cover more than the usual SMP: apparently this has to do with my r.(s,t)Kthere [forget about A ). I would be obliged if you} tell me what it is 7A all about, and whether similar results are pertinent to the general Markov process? For example, are the three Corollaries on p. 17t automatically true? Not. *olt ,W;/, ( I /u "el \ A r6 6,,,zc April 8, 2003 URGENT: Prof. John B. Walsh 4/9/2003 Please tell me why in the definition of FT for any function T from Omega to [0 oo) we need to put Po+ into it? If we use (7) on p. 16. with t=0. we get (0 I remember vaguely this "problem". What is it? By the way I assume you need FT_ in your part? I have begun my corrections. May have to skip some scribbled on my copy twenty years ago, no longer legible or understandable, tant pis. / 6o7 6c4~^ b*k 6 ^ ^ ( April 15, 2003 John: I am shocked that you say my original formula on p. 25 turns out to be OK. The correction of b for a was made probably after somebody in my class told me, but I must have decked it then. Now I am incapable of doing so (having tried for several days): would you be so good to show me how to get it from Doob's inequality I printed for the Upcrossing of a SUBmartingale? I am really curious. As you suggested I even looked at the unique copy of my 1st ed. of Course: there are several forms of such crossing number estimates but none of them can be easily converted to get what I used (foolishly) on p. 25. Please show me. I shall rewa'fd you by giving you a copy of the 3rd ed. incl. measure and integral. I am around p.80 now and found another correction in my copy that I can no longer check. I'll try it once more later and may have to either ignore it entirely or ask you to take a look  itis east stuff. Do you have a good grad student whom I can hire to check cer tain places? I used to have but only one or two were reliable Now of course none. By the way I got a letter from Doob (about Meyer  did you go to his memorial ?) He is 93. ___. ck^r c Q"c ) ,82 z z 4C)74 Urgent to Prof. Walsh John: Please take a look at Prop. 8 on p. 84, and see rgpp. 5 on p. 82 for f. I made some correction there but do not see it now. Can you see) Please reply  I am about to finish my parts and read what you sent me recently about the chains. One thing: except for stickiness and such fancy doing, my old M. C. book (2nd ed.) has all the examples you discuss there and may be a better reference for the average reader. I already told you the example of all qi=o is not Levy's (for once his intuiJt4tion failed, as I reported in my book for one (more?) pound of Canadian bacon. See 1il. cit. oov, AyqA24 42 6"),2 (DO;jL Lojt P. Ks) April 26, 2003 < ^ 4/29/2003 12:05 PM FROM: Fax TO: 101594516508570208 PAGE: 001 OF 001 Hi, Kai Lai, Propositions 5 and 8 look ok to me. I suppose you could mention in Prop. 8 that (10) implies that f is superaveraging. Can you be more specific: what in particular should I be looking for? Cheers, John Sorry to trouble you. On my teaching copy I scribbled under (15) on p. 85 (not 84 on my last fax) and added things on the last line f = liml lim ... J k n Can you see if it is OK?? I really can't read it but will leave it alone with your support, or make changes if necessary. It is too late to "improve" what is printed there, I just want to make sure it is not wrong. Please say. Good news for your discussion of Markov chains, see p.56, No. 3 with page reference. Forgot if I asked you if you need all those universally measur able stuff? Where? I remember that came from Meyer: Hunt only introduced "nearly Borel". But in your generalization what do you need? KdcA ^ D to John B. Walsh May 2, 2003 John: I caught it, tant pis. Did you realize that on p.85 where I asked you to check t>t I used the fact that min(fk) is exc.? Of (see my GOOD Index) course I proved it, but only much later and I went out of my way to say that it is a hard nut. Maybe you know an easier proof??? So my 20year old correction will stand: use fk instead. Unless you know a faster way. By the way I have''braching point on p. 9. Maybe you should use A P0 in your example of such a point. Have you seen my (unique) paper with Glover on left continuous MP (refereed by Andrd). It would be nice if you can use any of the results there as I recall some are rather tricky. Some time ago I ask ed him if he had any application. je said no, but you knww he is now some kind of "dean"  Art was provost. Prof. Walsh Please look at p.221 top. I scribbled there: It follows that the max pr. holds without requiring mu to have compact support by considering mulK (mu restricted to K??) Does it mean anything? I will send the corrected galley some 250 pages to you, but please make a revised ( by you) copy and mail one copy to me, keep another with you to be mailed together with your MS to Ina. Is it OK? Surely your good secretarial support can do the machinework for you. I can't do it myself. The French potential theory sans P has something like supermedian if not excessive. They do not have rightcon tinuous paths. So how do they prove that the min of two of their super... functions are super.. too? Is this true for positive superharmonic functions??? 22. Z6<+ Vancouver, 5/5/03 Hi, Kai Lai, Here is a section on stochastic integrals and local time, which I'll send you while I'm slaving away on the duality stuff. (Either I'm missing a few pages of those Paris lecture notes, or I finished the lectures before I finished the subject, leaving a few questions dangling.) And of course I keep thinking of more things which should go there. Anyway I have to figure how to prove a couple of extra things which are no longer as clear to me as they once were. In particular, why did I think it was completely obvious that "semipolar implies polar" was equivalent to "fine topology equals cofine topology except on a small set"? Anyway, I doubt that the enclosed section on stochastic integrals and local time will make it into the book, unless you can think of some use for local time. If you have any ideas, let me know. There are lots of uses, but not a lot of time, so we need something short and sweet, but still important. I suupose we could always use it to construct those examples of Markov chains with one or all states unstable, of course, but that wouldn't fit in so well with the rest of the chapter, and it's not enough by itself to justify all that work. But it's mainly for your interest. You said I could have five pages to develop stochastic integrals. That was enough for the L2 case, but by the time I got Ito's formula for local time, it took eight. I'm bored always repeating the usual construction, so I gave a different one, constructing the integral directly as a martingale. The main existence theorem is Theorem 1.5, which states that a couple of Hilbert spaces are isometric. It simultaneously gives the existence of integrals, the L2 bounds, and Ito's representation theorem, although I didn't prove the latter. Not good pedagogy, I'm sure, but I think it's cute. Maybe I'll send a zeroth draft of the hpath/duality chapters to you shortly, incomplete as they are. If nothing else, it ought to put your mind at ease about the connection of this stuff with duality, since I have a nice story about Markov processes in duality and electrons and holes in semiconductors. Damned if the story doesn't even make a couple of the balayage formulas look more reasonable. (e.g. why the equilibrium measure of a set sits on the union of the set and the points coregularnot regular for the set.) You can't see that in your case, by the way, since you assume that u(x, y) is symmetric in x and y, so that the process is selfdual and coregular equals regular. Cheers, h^ c^+^H^ k [2t a t h o aOfc a s5.2. SMPta r dr inml y .4 221 We am aow ready to etal anotder meaor priaci kinow a the Maaris Max himx PriJkpfk en flow. For y iteW mIeasre p supported bytfc iK, we hae (M) 4 pA(x) sop Ux)(M) ThI prairie a be ittivdy obvious to a phyiS, mir it ys that the pommel inu led by a dYarp.is gmea whI th dawp li Yet is poo s mas to dpm a m eort otdof dlity m pria a the diribaM of the ra 'Thib to bow that plyrical proin w rr an ihert dualiy ll d byw the alfdmtim Thmm The mifiM pk*f(f (M) AMb rider (S (T,) m (UA Prf. 'T hi mhiag to dm ifte rid i mbr of((M) is ilaht so w my ppmu it fit mid eql to M. For g >0 dd~a t ie B {xe BUp(Ax)s M +s). (12) Sinm Up is ewi udhr (UI), it is abl coetiodeom (Corolay 1 to Theirom I ofp.5) amd so B is aly dokd. The e ooaliaity of Up aso 7. ii that K c lr (whyr) Thifr wre wive by Theors lad Fobimi: k Cev9 PUp(x) fPas(x, y)M fP.(y, xO(dy) '(y .,x <,, ) Up( x) (13) 3 beMae l c K d f dbr ye a K, P(y,.x) ( a(Y .4tivit ymn e . S7O PA(x,) hbas nppo in I by TlhBor 2 dA .4, aMe *i adly domd. Hmes we have P /Upx) Pa(xdy)Up(y) up Up(j) ; M + 114) Putting (13) ad (14) teher we concede that Up M ies m i aurbitsry. [* e arpmaet ledimng to (13) is piiem and maoedd bow. CaSiey. For ny #in* mIrmee p aawk da L c r, we tah U PPUp. (IS) _ _. ,. .f ,1 , fftea Aoapl6L fMe ^o Lo ^ t4~ Re I Su> "Iel 'e de7CCa3er^,c~ C SU".L F!s Prof. John B. Walsh May 11, 2003 Sent you a copy of the page where (M) is proved, with marginal notes. As far as I can read it now, after over 20 years, I do not know why it was assumed that the measure mu has compact support. Can you duszover where it might have been needed?? I have now looked over five books including Brelot, Rao, Landkov, ... Nowhere except in a first discussion in Brelot where he began with a bounded open set (I do not know why) do any of them specify a compact support. In case you ahve any such book, such as Doob's which I did not check, you may try to find any hint for my past doing. Anyway if you can guaran tee that the proof given in my text goes through for any sup port of mu, I will just delete that stupid reduddant condi tion there and leave all the rest alone. NO SWEAT. Just in case I really needed that stupid condition for the given proof there, I will have to add a few lines to get the general case (more or less as you showed). That woyld be pain in the neck. Now I wonder how many other such stupiditoes Prof. Walsh May ll, 2003 John: My last fax may have confused you. I read my proof on that page and did not see ee*" WHERE I used the compactness of the support of mu. If you agree please fax and I'll just delete that stupid unwanted assumption. No Sweat. If you do find that I need compact K there I must add a few lines as you showed and state the result in the general case. Worse luck. May 13: I am waiting for your reply Jo the above to finish my part. You will have the corrected galley before you finish yours, and I expect various references (if need be) to s part if you can insert them somehow in the t~y part. But that would not be absolutely necessary: The way out is for us to say in a new Preface (abs. nec. this!) ~csy something about the connectionandlackthereof so that the good read er is.at least warned. I am Mailing you the first six pages of the stuff on .chains with some marginal notes. Nothing substantive and some may be outofdate. I have only glanced at the Ito stuff you sent. One way out is to put it in an Appendix! Nowadays it is so popular with the financiers that any dropping of the name may sell a few more copies (to rich brokers). One historical note you may not know: Doob told me it was he OW) who put the martingale connection in the integral. I was IR W ln proofread Ito's original memoir in the AMS and know that no martingale made its appearance. It is tries probable that. had the idea without spelling it out. So what you wrote there may re ~re a footnote of "historical perspective". Read also in my book with Ruth my presentation of McKean's way of doing something with max Bt but I forgetAIIt ab oJ local 6:me, (?) This graph, originally from the Washington Post purports to compare the income of doctors to other professionals from 19391976. It surely conveys the impression that doctors incomes increased about linearly, with some slowing down in the later years. But, the years have large gaps at the beginning, and go to yearly values at the end. 5/15/2003 10:38 AM FROM: Fax TO: 101594516508570208 PAGE: 001 OF 002 Vancouver, May 14 Hi, Kai Lai, This is the second day of trying to get this to you. Your machine seems to be sulking, and refuses to talk to mine. And the fax is getting longer hourly, as latebreaking news arrives. (The last paragraphs have changed three times already as I find counterexamples to the previous statements. Tomorrow... Who knows? Classes are over here and my exams are finished, and I'm not getting into the depart ment every day. So I may take a few days to reply to your faxes. I've been working on the final chapter of my partpotential theory under duality and it turns out I'd left my Paris lecture notes unfinished, so I've had to do some hard work to show that under the proper (strong!) conditions, for an excessive function h, that (1) the sup of potentials dominated by h on A and having compact support in A (2) the inf of excessive functions which dominate h on a neighborhood of A and (3) PAh are all the same. And when they are different. It all has to do with hypothesis (B) and quasileft continuity. This is admittedly dotting the i's and crossing the t's, but I'm feeling pretty good about it. The annoying thing is that I have no idea if all or any of this is new, but as the key tool is the exterior reduite, it might be. (I don't even know if I invented the exterior reduite, but with that name, probably not. Do you know? Have you ever seen it elsewhere?) You'll be glad to know that I needed Choquet capacities and could duck half the proof by referring to Prop. 3 and Lemma 4 in 3.3. By the way, this chapter is overlapping your chapter 5, so I said that for comparison, we would do the same thing by more classical methods there. Next in line is the "semipolar = polar" problem. As you say, it is a symmetry question. Basically, it is equivalent to "fine topology = cofine topology." That's true in your setting, of course, since you have a symmetric potential density. I have to work out a few details about exceptional sets to get things to look pretty, but it should come around shortly. I hope. It's fascinating that Doob put the martingales into Ito integrals! How did Ito do it without them? He must have proved a special case of the martingale maximal theorem to get the convergence of the integrals of simple functions. I'm cool with any plan to make use of local time, etc., but I wouldn't expect to fool even rich brokers into wading thru chapters on Hunt, Ray, and dual processes in order to get to the definition of local time! McKean does have a nice stochastic integral derivation of local time in his booksubstantially what I gave youand he has a clever way of constructing the stochastic integral itself, using an exponential maximal inequality. That's probably what you asked about. But in the end I don't like his method as well as Ito's. The questions you asked in the May 7th fax require a trip to the library, unless by happenstance I have the answer in a book at home. It'll probably take a few days, and in fact, my plan is to try to finish my notes, send them out so that someone will have a chance to tell me about some of the most obvious errors and misprints, and maybe help with the attributions. (Mostly missing at the minute because, while I know the math, I no longer remember where I learned it.) After that's done, I'll need to spend some time in 5/15/2003 10:33 AM FROM: Fax TO: 101594516508570208 PAGE: 002 OF 002 the library looking things up, and I can get to your historical questions then. Keep asking, but don't be surprised if I don't answer. Maybe you should make a list, and give it to me when that time comes. For example, I don't remember what Brelot's hypotheses were, but if he assumed that his cone consisted of lowersemicontinuous functions, that would do it. Or did he simply assume the cone was closed under mins? Doob states the maximal principle for Borel sets, not compacts. (Pg. 67 of his book, for Brownian motion.) His statement is more general, and he calls it the Domination Principle: you want part (a) with v = constant. I imagine that the reason that compacts usually enter the statement is that in axiomatic potential theory, one often wants to use it as a hypothesis, not a conclusion"Let C be a cone of potentials satisfying the complete maximum principle"and it is enough that it hold for compacts. And of course, if the set is not compact, you have to define what you mean by support. I think your proof is OK with a trivial modification: you have to replace supp(p) C K by "u sits on K" or "~(Kc) = 0" since if K is not compact, it can carry p without containing its (closed) support. The only place I can see that compactness might enter is your conclusion that K C B', but that's OK for any K, as long as it contains no branching points... (chuckle, chuckle) I just noticed that by taking h = 1 the maximum principle drops out of a theorem I just proved (namely that (1), (2) and (3) are the same if you have Hyp (B), quasileft continuity AND if h is a regular excessive function.) I think that the complete maximum principle (having it hold for all bounded h, or at least all potentials) requires "semipolar = polar" but we'll see. (The condition is sufficient, since it implies that all excessive functions are regular. I just don't know if it's necessary.) By the way, in the setup I haveoh, well, the setup needs to be slightly strengthened quasileft continuity plus hypothesis (B) equals "the branching points are copolar and the cobranching points are polar." See how nice branching points can be? Cheers, John 9~Qr am Prof. Walsh Friday 5/15. 2003 John: Your reply is not Clear to me. On p. 221, line 3, I deleted "compact" so that K is any support of mu. This means Mu(KC)=O. Some people (not Doob in his 1984 book, where wrote "supports" but FORGOT to say what they are.) like Dellacherir"Meyer define "THE support" to be the complement of Kzix~~t xi~f xkiXl un mxxxamztyxmayxixfxamxxEHixxxKi the union of all open sets U such that mu(U) = 0. Is this your definition, if not what is it? Now on p. 221, after (13), I wrote "because j1 CK", hence that support \& is a subset of Br (see the line above (13) where I wrote "why"  because I think of right continuity or if you will fine continuity. Hence any path starting from that particular A will hit B at once and so by definition of B in (12), 64O. as in text. 1 asa Ra 7 do not see why my original i LC is wrong although I can change it to "'K supports L ". Please explain. There are many different "supports" of mu, it seems to me the one defined by DelMey in their big bookT(pkease check) is the smmalest of all supports. Is this true? If we denote tha tky \ .taienfty (arbitrary) support K must satisfy LFQ as I wrote, and the supremun of U,.) over 11 must be < the supremum over K as written on the righthand side of (M) on p. 221. ISt M0Bie8 I am really confused. )ypCl(i by jS fP 4 7L! 2fA JPZq~)S I~ Prof. John Walsh: Nicht Urgent May 28, 2003 I got your fax about htransform etc. I do not recall any connexion between this and reversing. In my book with Zhaotal ready in second printing/edition! there is a lot of hBM there! In a Lip domain and using the Martin kernel K(.,z) as h. we prove that the transform is nothing but conditioning on the first exit PLACE X(T). If you can generalize this it may even be new. But I quit. aCt The rest is for Yuke. Dear Yuke, Glad to learn that Kevin (whom I remember seeing when ar ound three of age) WRITES and has been to Guatemala. We were there some twenty or twenty five years ago and I like it very much except the Guatemala City. Antigua is fabulous and ifs GENUINE coffee is one of the two or three best in the world, not available here (all fake or else). Atitilan (maybe there is another t somewhere) was praised by Al4pus Huxley as "even" more beautiful than Como. I do not agree but that may be due to the fact that I (we) have been to Como Some ten times and to Atitilan only once in a tour. See wha Kevin WEITES wav put my comment versus Alious in there. I am hoping to go to Guatemala again .... Best regards to you and your sons. Wl cPro Walsh June 5, 2003 I spent 15 minutes looking for your definition of "minimal excessive function" and did not find it. I bet there is none. Of course I do not need it. One of the stupidest custom of typing/printing is to omit the page numbers on certain "beginning" sheets  I never under stand or guess the PRACTICAL reason for this: maybe your good expe rienced st~hographer can tell you? By the way, is "stenographer" in vogue in good old British territory? 6?* . Shizuo gave me most of his Imperial College shittypaperg reprints containing his doings for the Nevaliva theory (??) that becomes a treatment of the boundary value problem (your teacher was a fast followers c. 1944  before the bomb). Not only I read it but I reprinted a whole page in my renowned Green, Brown, [NOTABEBE: do not let your good stenographer omit, by stupid cus toms the important comma after Brown' Some 93 percent of my correspondents omitted it probably never saw it.] and Probabili ty, second Edition, Mb w come I never saw any "holes" in your scanning? Take another look and point out to me where the holes are??? Did your teacher, not to mention Andr4 or Gil, see those holes? I found an unmailet envelope contain the pages on chains with a few unim rtant marginal notes by me. When I was told by the post office (its 800 number telephone) that Canada costs 85V (China or France only 80/) it was not mailed because I do not have 85/ stamp. I will ask Isolde mail it sometime. You (We) are contracted to turn in the MS in June. Just phone Ida. She seems to like you a lot. (66+ 23 TRANSMISSION VERIFICATION REPORT TIME NAME FAX TEL .06/04/2003 14:05 DATE,TIME FAX NO./NAME DURATION PAGE(S) RESULT MODE 06/04 14:04 101083412016175975 00:00:40 01 OK STANDARD ECM ( 0) 2 ,z ( It& .0 /.4' ^ ..c,. "'7.  ^ i0. l/ 7 ^ nS ^ i7 ^ ^ch sw94 Prof. Walsh June 8, 2003 John: Please show am the proof that I.yl1 is minimal excessive. If you have Notes as I did, explain in engineering detail how a semi conductor works the holes. host readers would not know. Call Ina about the delivery of our MS. On second thought, it would serve the reader better if you write a preface to the new edition, say what you want, and let me make some suggestions if'any. This will be printed after (or before?) my old preface (with a singQle misprint corrected as shown). The reader may read them in either order or not. I know most American students do not read prefaces (did you: but the "experts" may. Too bad Andre cannot, Doob might (if you sern him a copy of your preface), and I suspect Gil won't AFI assume he is alive but nobody told me). You should add (tems to the Index, that is necessary service and as you know really useful for any such book. When the galley comes, I will proofread my part, and scan yours. There will be misprints and mistakes. There were too many small misprints in the old text  my fault but Wu Rong simply was not up to it. If you have/ good studentkyou may employ one to help you. As I said the deadline may be postponed some, but "hopefully" not too long. Againteak to Ina. Good luck! I will mail you the few pages on chains, not important. What result by Pittenger you mentioned? Cci C~oi^^e^^ 0~ 6/11/2003 6:00 PM FROM: Fax TO: 101594516508570208 PAGE: 001 OF 003 Vancouver,June 6, 2003 Hi, Kai Lai, By coincidence, I went into the department today. Don't come to expect rapid answers, it'll probably never happen again. No postdatingyou got it shortly after I sent it. I'm a nightowl. The thought of getting up at 3 or 4 am is frightening, but I have no trouble staying up that late. Great that you found the reference to the MMPyou really have to know what you're talking about to recognize it, but it's there, and the invariant field is just the right one. (Damn, I should have called attention to that in a couple places in my part, since I did introduce the invariant field, and didn't make much use of it afterwards.) Yes, by all means write something about it. Old friends first: I don't know Dellacherie's address or phone, but can probably find it. I can't remember complaining that Meyer preferred Claude to his other students; it is clear now (and probably was then) that he was the best of them all. I'd probably have been talking about Maisonneuve, who wasisa good friend. The inventor of the predictable sigma field (and perhaps the predictable stopping time) was Catherine Dol6ans, now Catherine Dol6ansDade. Andr6 had divided them into accessible and totally inaccessible times. Andr6 was always very scrupulous to credit her. (By the way, Doob complained bitterly about the French stealing Illinois' best algebraist when she married Everett Dade and moved to Strasbourg. I didn't ask Andr6 about the American's stealing French probabilists when they moved back to Illinois.) About the firstname lastname question with Meyer: don't forget that I arrived there the year after Les Jours de Mai. The studentteacher relation had changed entirely, the profs showed up in jeans instead of a 3piece suit, and the grad students tutoi'd them as a matter of course. I think you were there slightly earlier, when the rule was as "vous." As a matter of fact, I never did get comfortable calling anyone "vous" in French, a great source of embarrassment to Joke at least once, when I addressed a venerable 80year old matriarch as "tu." I got a reply from Ina. A bit of a problem: I'd asked how much space I had. She replied that she hoped I could do it in 110 pages. I replied 1 was up to 156 and counting. I just got a reply to that one, saying she hoped I could do the remaining notes in less than 13 pages. (That was a great solace, I was afraid I'd have to drop my section "Death and Transfiguration: a Fireside Chat.") That may not be a problem: I find I have a great desire to keep it short and finish with this. But it also means that I won't do anything on Brownian motion, a pity since after all that work, I should apply it to something concrete. And I'll have to avoid brilliant second thoughts which just have to be included. 6/11/2003 6:00 PM FROM: Fax TO: 101594516508570208 PAGE: 002 OF 003 Forward to the Past! (1) About the MMP: I can't remember the aha! moment, but I am sure we invented it. (And so we're responsible for the name, too. No regrets permitted. Actually, I like itits a dig at mathematical names: the strong this, the weak that...always the extremes, why not the moderate something? But then, I suspect it was my coinage, so I have to like it.) Anyway, I certainly didn't learn it from the literature. If you think about it, there's little cause for itcertainly Hunt didn't have a chance, since with quasileft continuity, the process is continuous at any predictable time, so the moderate Markov property is part of the strong Markov property. About the only place you'd be likely to run into it would be in Markov Chains, where the first infinity is predictable and the SMP certainly doesn't hold there. But in that case, you don't need a general theory to deal with it. So I think we're pretty safe there. (2) Unilateral limits: my take on this is, when in doubt, make a RayKnight compact ification. (The Ray process IS the most general Markov process.) Then the left and right limits will always exist. In the Ray topology, the left continuous process is always moder ately Markov. (Which means, by your definition, that a Ray process is both strongly and moderately Markov.) Under duality, both processes are moderately Markov. (Thm 5.6.1. I suspect that's new, tho it certainly shouldn't be. But I haven't followed the literature, so I don't think I'll make any claims. Actually, I have the same problem with a number of results.) Now I have some historical questionsand I'll probably have a lot in the near future, as I try to do the notes on my section. (If I don't bail out entirely.) (a) History of reversal. Kolmogorov, of course, but his reverse wasn't homogeneous. The first real reversal from a random time that I know of is Hunt in 1960, with his approximate Markov chains, reversed at last exit times. This is surely where Nagasawa, Kunita, Watanabe, et al. got their ideas. It certainly influenced me. However, there might well have been some reversal from a firsthitting time beforedid you do it for Markov chains from a first infinity, or something like that? This reversal stuff had been floating around for a long time, but when did it first get written down, and who did it? (b) What about the doublereversal arguments? Did you ever use it to discover a theorem, prior to our article, natch? Lots of people did, I think; it was one of those things that were obviously true, which you couldn't use in a published work since nobody had made it rigorous. (c) I proved a bunch of theorems about Markov chain paths in what I sent you, and I know some of them go back to specific individuals. I remember you mentioning Donald Austin, and I know that one of them goes back to Art Pittenger. Do you think you could go thru that section and tell me. (Yes, I know that I can get it from your book, and I will if I have to, but it's be easier for me if you'd point out any of the results which were 6/11/2)03 6:00 PM FROM: Fax TO: 101594516508570208 PAGE: 003 OF 003 reckoned to be brilliant work at the time.) (d) Whn you're tracing the MMP, don't forget that lastexit decompositions are a form of Markov property. Might be worth mentioning. (Hey, its a firsthitting decomposi tion for the reverse. At the moment, I'm only devoting one sentence to it, but I wouldn't mind saying more.) Cheers, John Prof. Walsh June 12, 2003 Did you see the footnote on p.44 of my Random Time? I did not sayso if'our paper to ^iel Andre. L'etht:fGCest mo):proved a reverse Markov process has stationary transition  Meyer failed even to do this in his (later withdrawn but I have a copy) Seminaire cited in the footnote. Read it. My reverse is from the first infinity of a chain, and that is the model in our article. I need some smoothness of the az i e which you supplied witr triple exponential prolonging. I did not vaunt these matters in my Notes because MC was only mentioned in simple examples (Paul Levy chided me in his review of my Course for not treating M,C ). History is a tricky thing  better keep mum when you are not sure. Assuming you have a copy of the Proc. ICM at Nice (where I ate boursin f@r the first time, grace a Yoke [Yuke? maybe Joke in Dutch: ask her please), the history of MMP is told in style and my prediction regarding its prevalence now seems accurate. In 1969 it fd called "moderately strong", the precise name may welmIeade its appearance in my talk at Nice.. "f you want a reprint I can send one. Regarding double reversing of course I did it some where but now forget where. Be careful: one needs a primary condition something like the reversal time be finite or somethinge In a conversation with that Jap in Zurich I forgot to say so and he, japlikes jumped on it. You did not understand what I recalled about Meyer + Claude: your complaint was that Meyer preferred C's counterexample to yours. I found that error in C's book. Repeat: for any remembrance of the past, DON'T say it unless you can find the record in print. Yushkewitz told me a story about "Eugene claiming some result not his, with kudos from his coterie, but the true story is in print. /If you contact Art Pw, let me know. My regrence on p. 74 of Lectures is faulty: that book was published before 1969 and does not contain the name MMP, only the result written by / Meyer. Hope this ends our chat about history. ,knee./X ;Z 33I Additioi'be Bibliography K. L. Chung ~l] Probability method in potential theory. Proc. Conf. Poten tial Theory (1987), Lecture Notes in Math. 1344, Springer Berlin. [12] Green, Brown, and Probability Brownian Motion on the Lines World Scientific Publishing Co. Singapore 2002. J. L. Doob [ Classical Potential Theory and Its Probabilistic Counter part. Springer 1984 r91 On the boundary theory for Markov chains, Acta Math., 110 (1963),p.1977; 115,(1966, p. 111163. [10) Bpoundary behavior of Markov chains and its contributions to general processes. Proc. International Congress of Mathe maticians, 1970~ p. 449505 (with one picture). published by GauthierVillards 1971. John Walsh: June 13, 2003 13 and Friday Please make sure the above additional Bibligrpahy is print red, at an appropriate place. Other Bibliograohy needed for your part may be added or (easier?) printed where your part ends? Up / to you but tell Ina our choice. The format has been changed as I v d added "p." wkich was omitted in the old edition. Now please figure out an optimal mode to insert the following additional Note (as we discussed in faxes),an Vpper. of )(P ,k be e'th fe Notabene: the nane "Moderate Markov Property" made its first appearance in Chung [10] as far as we can I trace it. Atkkxgk Indeed it was fortuitously observed _____6 during uwxx pageproof maxxXKXXza~i of Chung 19]3 pp35, and the lefthand approach is liA&o re for crossing the barrie ' V ~bret eri Later when Walsh saw its iR~ RRR*x natural  4k rlME wIwhen time is reversed (right becoming left) in Chung and Walsh r]de iag ardOje ilml 4stnuA rwa name "Moderate Markov Property" was introduced. Since it is much in evidence in thi new diten aet this story be addcL tod here, )A nevi chpers is '^*4vr4f. pleasure ai 4ellv )1"Ti;t, JUw~o t an /D P;Z 3' S f K. L. AdditionSt Bibliography K. L. Chung 01] Probability method in potential theory. Proc. Conf. Poten tial Theory (1987), Lecture Notes in Math. 1344. Springer Berlin. [121 Green. Brown and Probability Brownian Motion on the Line, World Scientific Publishing Co. Singapore 2002. J. L. Doob 4 Classical Potential Theory and Its Probabilistic Counter part. Springer 1984 F91 On tLe boundary theory for Markov chains, Acta Math.. 110 (196*),p.1977; 115.(1966, p. 111163. [101 Bpoundary behavior of Markov chains and its contributions to general processes, Proc. International Congress of Mathe maticians, 1970. p. 449505 (with one picture). published by GauthierVillarde 1971. Additionirt~ Bibliography K. L. Chung 1] Probability method in potential theory. Proc. Conf. Poten tial Theory (1987), Lecture Notes in Math. 1344, Springer Berlin. [12] Green, Brown, and Probability Brownian Motion on the Line, World Scientific Publishing Co. Singapore 2002. J. L. Doob [4] Classical Potential Theory and Its Probabilistic Counter part. Springer 1984 [9] On the boundary theory for Markov chains, Acta Math., 110 (1963),p.1977; 115,(1966? p. 111163. [10] Bpoundary behavior of Markov chains and its contributions to general processes, Proc. International Congress of Mathe maticians, 1970, p. 449505 (with one picture). published by GauthierVillards 1971. Additio~Sbt Bibliography K. L. Chung 1)3 Probability method in potential theory. Proc. Conf. Poten tial Theory (1987). Lecture Notes in Math. 1344, Springer Berlin. 1121 Green. Brown, and Probability Brownian Motion on the Line, World Scientific Publishing Co. Singapore 2002. J. L. Doob 1 Classical Potential Theory and Its Probabilistic Counter part. Springer 1984 r91 On t.e boundary theory for Markov chains, Acta Math., 110 (1963),p.1977; 115.(19661 p. 111163. C101 Bpoundary behavior of Markov chains and its contributions to general processes, Proc. International Congress of Mathe maticians, 197l. P. 449*505 (with one picture). published by GauthierVillards 191. /6?/SA / 1 Addit o Bibliography Addition'** Bibliography (2dZ  K. L. Chung I / Probability method in potential theory. Proc. Conf. Poten '1 dtial Theory (1987), Lecture Notes in Math. 1344, Springer Berlin. 1 12] Green, Brown, and Probability Brownian Motion on the Line, World Scientific Publishing Co. Singapore 2002. J. L. Doob [41 Classical Potential Theory and Its Probabilistic Counter part. Springer 1984 [91 On the boundary theory for Markov chains, Acta Math., 110 (1963),P.lA77; 115.(1966? p. 111163. I~J( [10] Bloundary behavior of Markov chains and its contributions S to general processes, Proc. International Congress of Mathe maticians. 197,. P. 449505 (with one picture). published by GauthierVillards 1971. / by autieVilars 9V1 I T.Yh 4 %( witL Af ia h 6 ' ^**/^^lY. h q A) "A '.B Notabene. The name "Moderate Markov Property" made its first entrance in Chung [101. It was fortuitously observed during galleyproofing of Chun4[9;P.35],when the boundary was crossed at the "first infinity" time. John B. Walsh saw its natural pertinence when TIME is reversed, 4" Chung and Walsh [l11]where it was called "moderately strong". Since this left handed variety of Markovian behavior is munh in evidence in this new edition, its L ..,,:. i: i.,. ere for the pleasure of the company . June 15, 2003 John: This is my "best effort". Any suggestions are welcome but please fax me soon. You write well, generally speaking. Notabene must be printed as is: you knww what it means in Latin. K fl1^ e^U 4 ^w^ ^^t. III ...... 1 6 S tL W"j 06p7c^ ye &_ .. Prof. Walsh June 16, 20'3 John: Please discard the fax I sent yesterday. une Insert this revised one on my p.235. The Notabene (must print as written, NOT N. B..) may be inserted where you choose, but let me know. This is my final act. Ciao. Additiorgr*G Bibliography K. L. Chung  l] Probability method in potential theory. Proc. Conf. Poten tial Theory (1987), Lecture Notes in Math. 1344, Springer Berlin. [12] Green, Brown, and Probability Brownian Motion on the Line, World Scientific Publishing Co. Singapore 2002. J. L. Doob 14] 1 Classical Potential Theory and Its Probabilistic Counter part. Springer 1984 r91 On the boundary theory for Markov chains, Acta Math.. 110 (1963),p.1977; 115,(1966! p. 111163. [10] B1oundary behavior of Markov chains and its contributions to general processes, Proc. International Congress of Mathe maticians, 1970. p. 449505 (with one picture). published by GauthierVillares 19'1. 1 Notabene. The name "Moderate Markov Property" made its first entrance in Chung [10]. It was fortuitously observed during galley proof of Chung [9; P.35], when the boundary was crossed at theJ "first infinity" time. Later John B. Walsh saw its natural perti nence when TIME is reversed; see Chung and Walsh [1; Section 6], where it is clumsily called "moderately strong". Now that this lefthanded Markovian behavior is muchtevidence in the new edi tion its origin may be told e a M~MENTO. C xS) June 17, 2003 John: Since the book has "Time Symmetry" in its title, and reversing seems all over the places it is de rigueur to record some history about it. You should write it since you wrote most of it in the book [I only casually mentioned it.] But please send it before you send it to Ina, for a check. No we do not have to start with Kolmogoroff for a finite state chain which is a dual rather than reverse: read Feller and my M. C. book (my student Derman extended it to the countable case). READ first the footnote on p. 44 of my Random Time, to learn the status at that time, unless you 'ei. have Meyerts original articlewhich was withdrawn later That is PREhis jq/q tory. Then read the first page of our paper(196a Next, my Nice talk in the ICM proceedings. Read also pp. 271ff of my book on M. C., second edi tion. I had forgotten tYI note on Martin Boundary where I reversed from the first infinity. Here is the notyetrecorded history. I showed you the last bit and told you that I needed a density of the reversal time, the i(t) there which I saw also in your t4&. You supplied it by exponential three times. The MMP of the reverse was your great deed ... But we faxed about that already. After our paper Meyer wrote his "commentary" with some "improve ments"? That is for you to check. I was very kind to him in my footnote cited above. Yerf;n ts ? Let me know if you need any informations [French and Italian both have the plural] I assume you have my reprints referred above uch as the ICM article with a nice picture which you may wish to point to as branching etc. TIME is running short, so I am mailing this with the pages on M. C. you sent me months ago, but will also fax you a copy of this if need be. Post to your labourstrong country may be slow, I had heard our rightwing politicians say  as well as your health care persaipel? If you have ever been hospitalized you may know. Thank Heaven I have not been to a hospital yet since my eye operation. Cheers. tjei o ry Prof. Walsh, GENT June 18 or 2O3 John: An important reminder. You should write Historical Notes for TIME SYMMETRY viz. reverse/dual whenvyour part begins these doings, which (W to eva seem to me to occupy most of your space. Going back to Kolmogoroff is not enough: ftr the finite chain there is statiinary distribution (later ex tended to the countable case by MY student Dermans did you know?) and so there is "no problem". You obviously were not aware of the long article by Meyer cited on p. 44 of my RANDOM TIME book. I am sure you have a copy of that Seminaire because it was published when you were in Strasbourg (Moulinr). There he clearly stated the difficulty of getting a homogene ous reverse (i. e. with stationary transition): check this against my memory. The Japs cited on the first page of our article( 969), following Hunt (discrete time) reversed from some last exit time but I do not re member if they proved the homogeneity of the reverse. It is for you to check. The problem of true reversal, namely path by path, was at the time regarded either as "obvious" or "impossible and unnecessary". In my Lec tures I did not reverse (except Doob's from fixed t) so I said nothing. You should tell some and send it 82% me for proofing. Do not forget Meyer's article on ours which has some merits, and those other papers you wrote with Smythe et al (which I do not remember at all). There is a Russi at Cornell (in 1994 when I was there) who apparently did something .... Afftixxast If iTere is any literature after c. 1990, I know zero. There was an AMS conference on time reversing at Santa Cruz, where Azema (and his girl friend), Heinz Bauer (dead), Dynkin, Getoor, ... attended Unfortunately I do not know if there is any record of the talks. Anyway as usual everyone just talked about his own junk. I will mail you the few pages you sent Ke on chains with my own stamps but am not sure how much. In older days such mail was delivered with additional postage required. Let me know if this one "works". I am amazed how high the postage to nearby Vancouver. P. S. Example of lost history: neither Doob nor Hunt reversed zxxixx except as noted above, and yet Hunt spent a whole Me i No. II on "duality", followed by Blumenthal (Hunt's student) nd Getoor's book. Besides thdse holes you saw in the semic~nductirs e you can list some physical applications of your reversing in Part II. After all physicists think facilely of those doings from high heaven down to earth. and TALK Prof. Walsh June 16, 2003 John: I sent you a fax yesterday and today. The latter replaces the former. I have forgotten if I have already told you: (1) When I said Meyer preferred Claude to ..., I meant you. I found a mistake in Claude's book, you gave a counterexample but Meyer cited his  something like this, not well remembered. After all both Claude and you were then there and must have done similar things, "no problem". (2) It would be good to show a double reversal in your stuff. I did that in some place but remember you may need a preliminary condition such as the reversal time is finite. (3) In Part 2, where reversing time is a big deal (not in Part 1 where it is only casually mentioned), some solid history should be given in Notes or otherwise. What Kol. did c. 1930 is NOT reverse, it is dual. A GOOD history is given p.271 ff. my M. C. SecOnd edition [if your poor library does not have it, get one from somebody and READ IT.. I told you c. 1968/9 about my reversing the minimal chain from its first infinity and said I was unable to generalize it because I need a density denoted by li(t) there. You had the bright idea of prolonging the life time 3exponentially and the rest goes just as in my special case. C'est la vie. The MMP is your great contribution, is it by that essen tial limit? I forget. Need to read my footnote on p. 44 of my Random Time to appreciate this  maybe I have said so in another fax  wonder (f Meyer ever saw it. I did not send him a copy partly because he told me he was entirely out of math, and part ly I could not ask him to send me a can of pate (not goose, too heavy for me now) as payoff like your bacon. No kidding: your new part under Time Symmetry deserves a brief history (a la Bourbaki gxxz if you can. It should be "fun". 44/ Jlu,( Zef ~~~~~~^t^? Prof. Walsh June 19, 2003 FINAL Note et I John: I enjoy your expansive response but must warn you about the Notes you write (!) in your part about TIME SYMMETRY The book is for new readers who don't know a thing i4 e b.3^I Pe MN 1 r _ 4jsudlaL tL L ,~4r eicT TwoA%(2 b b M par phs would do. Who cares for all those "credits"/ I just read again the first page till the top of second page of ou an J=17r) article. DO READ IT, word by word if need be. 7 $*arted with IME mmetry and ewhi. the duality assumptions of the Japs. I was too kind to Meyer not to cite his unsuccessful account in the Seminaire, which is now cited on p. 44 of my poundofbacon book. You should find Meyer's artucle and perhaps quote him directly about the un solved problem of time homogeneity of the reverse (I forgot what he said so you must check it). Then we can simply say that problem is now solved here in Chapter ?, Section ?. Basta. YOU (not I) can just refer to our paper for more nittygritty history  who cares now? After we prove the path bypath reverse ofy a Hunt process, of course the reverse i left con tinuous but that is not enough and now you can "brag" about essen ial limits MMP (I already did it for you in N'ja eene ?). That is enough history fdr  the innocent reader. If you want to add more gossip that will be fine but I won't overdo it: Nobody understands nor cares. E. g. what was done in discrete time chain with a stationary distributionFis Neanderthal: we now think it was not Human. I forgot but just saw that I even citFe Ed Nelson's Chicago Ph. D. theses  I was there. Even Ray's paper is in my Bibliogra ohy in the'first ed, *f my MC book, 196o, and in my I~dex. I hope this is my TnaI effort to contribute to the new edition, except a very brief joint Preface, in which we can just say that t consists of two parts, etc. As usual that can be delayed until galleyask Ina. [My old Preface must be reprinted with one misprint corrected]J Postscript (June 29). If Ina wants to cut, here is an easy way. Just erase all those folksy style in the MS. suitable maybe if you talk in classroom but unnecessary and unhelful in a TEXTBOOK: GRUNDLEHRENT This was noticed right at the beginning of your MS but I did not comment on it since you are your own free agent  before Ina ... Ha! A good editor can do it for the inexperience author. Have you written a BOOK be fore (lecture notes do not count). I have written OVER a dozen. Prof. Walsh . une 20, 2003 John: cTt +4(; (1) Have you READ the artle about retourement du temps by Andre in Seminaire 1968. I found it and read it again. If you have not read it yet, do so before writing your history. I wouletstart it with a quote (in French) from our dead friend. .He was honest '( as others were not). (2) At least I hope you have read the FIRST page of our article but g do not stop unyil the top lines on the second page. I (Moi) would not give'the Japs [until their government makes formal confession and "apology" as the Germans'did to the Jews they will remain Japs to me? of course I do not extend this to some mathematicians I know)  repeat: IOULD NOT give those Japs mentioned on the first page too much credit because they be gan with DUALITY hypotheses (check those; I forgot). If you begin with duality what do you want to reverse for? Cf. Blumenthal and Getoor etc. Remember good old Gil not only had to assume duality but even further [Cartanlike, I forget] specialization for his No. Memoir  which we did not go through. Remember we went through a large portion of'his No. I I bet Gil did not assume/duality in his '(last) paper about Martin m~;W1 boundary, but that is for discrete time  pas de problem lA. WHY did he 1/t not do it in CONTINUOUS TIME? I did it in my Martin boundary note in continuous time (discrete space is no big deal) and that wa+MY MODEL for reversal. I think I told you so in an earlier fax with that density of the reversal time li(tyream glad to see it in y&ur MS: the key to . (3) It just occurred to me: in the continuous time I suppose we can also have a 'stationary distribution": e. g. for Brownian Motion it must be the BAgrLebesgue measure.: no problem with infinite measure as Derman had already done for countable chains [that was at that time quite .; something : even Feller put it in his book. The Russians never got it] 193 Now if there is a stationary distribution for a general "RayHuntFeller" process will it then be trivial to reverse as Kolmogoroff did (you cited it: but did you ever read his paper? I didO] I do not know the answer. (4) W&t really beats MEps I know of few if any good applications of reversal!l That woman in Cicinati once said she had some, but it was extremely special/. Do some if you can, not too late. The test of the pie. Of course I have'many great applications for "LAST EXIT TIME", even 'lectri city but that is in Part 1. Ciao. 5eAt ? 1/20 June 16, 2003 John: I sent you by fax my amended Bibiliography and Notabene, second version. Please have this inserted and tear the first version sent one day before. Your "history" of reversing is all wet. Kol's is not reverse: it is dual. I think it is given in Feller with extfon to the denumerable case ddne by my Ph. D. student Derman, in Feller's book. See s St 'm) n ,C. (REEAD p. 44 of my RANDOM TIME, the footnote. Did you? I was kind to AndrA there.ff you can find a copy of his WITHDRAWN artici le (I have a copy he would rather forget). He did not knww if a (Hunt) process reversed would still be homogeneous. See if he was honest to say so? There are perhaps "hints" of this in Hunt's (last Paer on Martin for DISCRETE time, but I do not remember whether he was able to show that the reversed chain (from a last exit time) s stationary transition amely "homogeneous". The Japs Nagasawa and others did reverse from ceratin "reversal" times specifically cited on p. 1 of our 1969 paper. But I do not remember if they proved in any manner the stationary transition of the reversed "portion". f you are interested you should ,reat it and tell me. As far as I know, the VER,FIRST TIME that a reversed process has stationary transition is given in MY book on Markov Chains, where the reversal time JS the FIRST infinity of the "mini f remal chain"  READ IT. It may be evlein the first edition if that'siotly one you can fnd.i The time reversal in conti nuous time was regarded either as. an "obvious" thing or as "who cares"? You are too young to know that. Your great S. 8M teacher asfar as I know, did not take it up. He did re 0"41 verse at a conatnt tl see MY Lectiures> but I do not recall +40 any other diddling by him, nor by the French schooluntil Yotr P t OUR (wow) paper. Read Meyer's rewrite which I eecall has s3mey merits (maybe you make use of these in Part 2  I did not see ~uYf"l& not read the details anyway). Were you at A the Santa Crrz confab where Azema. and 1i fe, .l. and dead Bauer were all there? I is W"W'y . KQ\I TRANSMISSION VERIFICATION REPORT TIME : 06/15/2003 06:96 NAME: FAX : TEL DATETIT 806/186 V:! FAX N"AME 10106340 0826 9581523 DURATION 00: 00: P (S) 01 T ANDARD CM I think it was supposed to be about REVERSING TIME, but as we know degenareted into all kinds *atuff. I did jiot remember if it has any record though it was partly supported by NSF, throgh my and a special grant. The German(s) complained about their shared dormitory rooms and asked me to protest. I did by telephone on the spot, spoke to XXX [Forget the known name]. You know how such things went. Hsu Pei (do you know his book in AMS??] ~gib me a ride,.... 3 Q(v 1' h stQ 'p8a n (s, ) t( wj (), /, \?, Cz) AC), /L E s F3, ( < c ' OLe'( s) b))c (j ~:)> c4 S/B~ai)~ z~(K6 S ELo z65) oi Sc,j~ ~~)) ~K ~i~6 9, ) I A)) 'Ir FI ) '4) G (e 1~) T(2~ IA ub) '1 'I L(bi) 1 i ayJ^


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