return

Celebratio Mathematica

Kai Lai Chung

Probability

Kai Lai Chung: A remembrance

by Ronald Getoor

Kai Lai Chung was an ori­gin­al, in the best sense of the mean­ing of that ex­pres­sion. He had a great en­thu­si­asm for math­em­at­ics and life that was con­ta­gious. Chung was one of the lead­ing prob­ab­il­ists of the second half of the twen­ti­eth cen­tury. He had strong opin­ions about math­em­at­ics and oth­er sub­jects, which he ex­pressed vig­or­ously and em­phat­ic­ally. He could en­liven any dis­cus­sion or meet­ing with his lively per­son­al­ity, en­thu­si­asm and ex­uber­ance.

He ap­pre­ci­ated go­ing to the best res­taur­ants, hav­ing fine wine and stay­ing at the best ho­tels. In 1970 we were both speak­ers at the In­ter­na­tion­al Con­gress of Math­em­aticians in Nice, and he took great de­light in kid­ding me that he was stay­ing in a five-star hotel while I was only in a four-star hotel. Ac­tu­ally, both ho­tels were quite old and very or­nate, and they clearly had seen bet­ter days. They were on the fam­ous Prom­en­ade des Anglais, which was a busy fume-filled noisy mo­tor­way at the time. The Medi­ter­ranean was just across the street from the ho­tels, the beach con­sisted of stones, and was not par­tic­u­larly in­vit­ing. So one af­ter­noon we de­cided to make an ex­cur­sion by hy­dro­plane to Cannes in or­der to see the fam­ous beaches and ho­tels at this well-known re­sort area. It al­most lived up to its repu­ta­tion. Any­one meet­ing Chung was not likely to soon for­get him.

He made fun­da­ment­al con­tri­bu­tions to many areas of prob­ab­il­ity the­ory, ran­ging from sums of in­de­pend­ent ran­dom vari­ables to prob­ab­il­ist­ic po­ten­tial the­ory. He was rig­or­ous and de­mand­ing in his ap­proach to math­em­at­ics. He had little pa­tience with ar­gu­ments that were in­com­plete. He could be­come par­tic­u­larly an­noyed with au­thors that slurred over cru­cial tech­nic­al de­tails. His bib­li­o­graphy lists more than 125 pa­pers and el­ev­en books. Moreover, he was an ex­cel­lent ex­pos­it­or. For ex­ample, his book Lec­tures from Markov Pro­cesses to Browni­an Mo­tion [5] is the best in­tro­duc­tion to the circle of ideas as­so­ci­ated with the names of Hunt, Mey­er and Dynkin. After it ap­peared, it was the book I re­com­men­ded for my stu­dents to be­gin read­ing. The little book Green, Brown and Prob­ab­il­ity [6] is a de­light. He took great pleas­ure and pride in his use of lan­guage. The prose in his books and pa­pers is of­ten el­eg­ant and en­ga­ging. I par­tic­u­larly en­joy his use of asides, es­pe­cially in his books.

I first met Kai Lai in the spring of 1955. It was dur­ing my first year as a Fine In­struct­or at Prin­ceton, and he gave a talk in Feller’s prob­ab­il­ity sem­in­ar. I be­lieve that he was talk­ing about Lévy’s work on chains all of whose states were in­stant­an­eous. This was quite new and some­what dis­turb­ing at the time. Evid­ently he had dis­cussed Lévy’s res­ults with Doob, and what I re­mem­ber is how he ex­citedly ex­claimed, “Doob kept ask­ing, ‘Where is the ran­dom vari­able?’” I must con­fess that I did not really un­der­stand the is­sue at the time. In any case, I was im­pressed by his en­thu­si­asm and ex­cite­ment. He cer­tainly was very dif­fer­ent from any of the Chinese that I had known pre­vi­ously. In pre­par­ing these re­marks, I real­ized that dur­ing my two years at Prin­ceton I don’t re­mem­ber any oth­er speak­er in Feller’s sem­in­ar, al­though there must have been one most weeks dur­ing the aca­dem­ic year. Yet I re­mem­ber Kai Lai in the sem­in­ar very vividly after nearly 55 years. This speaks to how im­pressed I must have been by Kai Lai.

In those years 1955–56, Feller was work­ing on his bound­ary the­ory, and I was try­ing to un­der­stand it in the hopes of find­ing a top­ic for re­search. I don’t re­mem­ber if I men­tioned this to Kai Lai, but a little later he wrote me sev­er­al times ask­ing ques­tion about Feller’s bound­ary-the­ory pa­pers. I answered to the best of my abil­ity. I had man­aged to work through the de­tails of Feller’s pa­pers, but I nev­er de­veloped a real feel­ing for the un­der­ly­ing is­sues.

I don’t re­call hav­ing much more con­tact with Kai Lai dur­ing the next few years, al­though I must have met him and spoken with him at meet­ings. Then, I spent the aca­dem­ic year 1964–65 vis­it­ing Stan­ford, and this is when I got to know Kai Lai. Ac­tu­ally, it was Sam Karlin who in­vited me to spend the year at Stan­ford, and made all the ar­range­ments. Kai Lai had come to Stan­ford in 1961, and as it turned out I spent a good deal more time talk­ing with Kai Lai than with Sam. Kai Lai had be­come in­ter­ested in po­ten­tial the­ory and its re­la­tion­ship with Markov pro­cesses. By then I had be­come in­ter­ested in Markov pro­cesses in the spir­it of Hunt, Dynkin and Mey­er. I lec­tured on this sub­ject for two quar­ters, cov­er­ing some of the top­ics that would ap­pear even­tu­ally in my 1968 book writ­ten jointly with Bob Blu­menth­al. Kai Lai sent sev­er­al of his stu­dents to the lec­tures. I re­mem­ber Naresh Jain and Art Pit­tenger. I also dis­covered a few years ago that Tom Kur­tz had at­ten­ded, but I must con­fess that I don’t re­mem­ber him be­ing in the class. Dur­ing this year at Stan­ford, I got to know Kai Lai, and we be­came good friends. We of­ten had lunch to­geth­er, usu­ally at a place called the Barn loc­ated in the Stan­ford shop­ping cen­ter. It was what would be called a food court these days, with vari­ous eth­nic-food places around the peri­phery, with tables in the cen­ter. It was a great place where one could linger and dis­cuss math­em­at­ics, with an ample sup­ply of pa­per nap­kins to scribble on. Oc­ca­sion­ally Sam Karlin would join us. At the time there was a joint prob­ab­il­ity/stat­ist­ics sem­in­ar between Stan­ford and Berke­ley that met once a month, al­tern­at­ing between the two cam­puses. When it was at Berke­ley and the top­ic was of in­terest to us, we would travel to Berke­ley to­geth­er. My re­col­lec­tion is that I usu­ally drove.

Kai Lai had in­vited Mar­cel Brelot to vis­it Stan­ford dur­ing the spring quarter of 1965 and lec­ture on clas­sic­al po­ten­tial the­ory. Both Kai Lai and I at­ten­ded Brelot’s lec­tures which, to the best of my re­col­lec­tion, fol­lowed more or less his well-known Élé­ments de la Théor­ie Classique du Po­ten­tiel [e1], which ap­peared in “Les cours de Sor­bonne” lec­ture notes. Dur­ing the first sev­er­al lec­tures the class room was com­pletely filled, but by the end of the quarter Kai Lai and I were the only at­tendees. It was a great op­por­tun­ity for me not only to get to know Kai Lai, but also to be­come friends with Brelot.

In 1966 I moved to San Diego, and after that Kai Lai and I ex­changed nu­mer­ous vis­its. Dur­ing his vis­its to San Diego, Kai Lai took great pleas­ure in vis­it­ing the San Diego Zoo. He was very in­ter­ested in an­im­als, and usu­ally tried to work-in a vis­it to the Zoo when he was in San Diego. He en­joyed hav­ing lunch on the patio of the La Valen­cia Hotel in down­town La Jolla, and we would try to have lunch there whenev­er he was in town. He would usu­ally have one and only one beer with his lunch.

In the sev­en­ties and eighties, we had an ex­ten­ded cor­res­pond­ence on vari­ous top­ics con­cern­ing Markov pro­cesses and re­lated sub­jects. This cor­res­pond­ence flour­ished es­pe­cially dur­ing the peri­od he was writ­ing his book, Lec­tures from Markov Pro­cesses to Browni­an Mo­tion [5]. Kai Lai’s let­ters of­ten ended with the ad­mon­i­tion “An­swer at once,” or just “An­swer,” or words to this ef­fect. I par­tic­u­larly en­joyed this ex­change of ideas, and I learned much in my at­tempts to an­swer some of his ques­tions. I am happy that he and I were able to col­lab­or­ate on a short pa­per, giv­ing the prob­ab­il­ist­ic mean­ing of the con­dens­er charge in clas­sic­al po­ten­tial the­ory, and that he chose to in­clude this pa­per in his volume of se­lec­ted works. Much later, after he re­tired, Kai Lai told me that he had de­cided to des­troy all of his cor­res­pond­ence, and after I re­tired I did the same.

In the late nineties, my daugh­ter was a gradu­ate stu­dent in Com­puter Sci­ence at Stan­ford. Kai Lai had re­tired by then, but when I vis­ited her I would call him, and of­ten he would in­vite me to his house in the af­ter­noon for a beer or tea, and con­ver­sa­tion. It was dur­ing one of these vis­its that he told me about des­troy­ing his cor­res­pond­ence. I was honored in 2000 that he came to San Diego and spoke at my re­tire­ment. Dur­ing that vis­it we had the op­por­tun­ity to have lunch on the patio of the La Valen­cia once again, and as be­fore he had one beer with lunch. This was the last time we met.

In read­ing his bio­graphy in the IMS Bul­let­in, I learned that he had wide-ran­ging and in­tim­ate know­ledge of op­era. I am also an op­era fan and in­ter­ested in op­era, but this nev­er came up in our dis­cus­sions. In fact the main top­ics I re­mem­ber dis­cuss­ing with him were math­em­at­ics, the polit­ics of math­em­at­ics and oth­er math­em­aticians.

Fi­nally let me say a few words about his math­em­at­ics. Here I shall make some gen­er­al re­marks on his work, since I already have made more tech­nic­al com­ments on sev­er­al of his pa­pers in the volume of his se­lec­ted works. My com­ments only ap­ply to that por­tion of his oeuvre with which I am most fa­mil­i­ar, roughly cor­res­pond­ing to the peri­od from the late six­ties to the early nineties. Al­though it is ob­vi­ously an over­sim­pli­fic­a­tion, I think that there are two closely re­lated ideas or, per­haps bet­ter, themes un­der­ly­ing some of his most im­port­ant work dur­ing this peri­od. Namely, the ideas of re­vers­ing the dir­ec­tion of time in a Markov pro­cess, and of us­ing last exit times.

Here and in the re­mainder of this dis­cus­sion, the term Markov pro­cess means more pre­cisely a con­tinu­ous-para­met­er tem­por­ally ho­mo­gen­eous Markov pro­cess on a reas­on­ably gen­er­al state space, for ex­ample, a loc­ally com­pact space with a count­able base or some gen­er­al­iz­a­tions of such spaces. The para­met­er set is usu­ally the non­neg­at­ive reals, and in ad­di­tion the pro­cess is as­sumed to be strong Markov with right-con­tinu­ous paths hav­ing left lim­its.

Since the defin­i­tion of the (simple) Markov prop­erty is sym­met­ric with re­spect to past and present, this prop­erty is pre­served if the dir­ec­tion of time is re­versed. But in gen­er­al, the tem­por­al ho­mo­gen­eity is not, nor is the strong Markov prop­erty, and clearly right con­tinu­ity with left lim­its be­comes left con­tinu­ity with right lim­its. One of the reas­ons for the in­terest in re­vers­ing a pro­cess is that, in Hunt’s work on du­al­ity the­ory for Markov pro­cesses, he proved the ex­ist­ence of a tem­por­ally ho­mo­gen­eous right-con­tinu­ous strong Markov dual pro­cess un­der hy­po­theses that were some­what opaque and dif­fi­cult to veri­fy. Later, in [e2] for ex­ample, the ex­ist­ence of such a dual pro­cess was taken as a hy­po­thes­is in de­vel­op­ing du­al­ity the­ory. It was clear that in some sense the dual pro­cess was just the ori­gin­al pro­cess with time re­versed. Thus, in ad­di­tion to the in­trins­ic in­terest of the pos­sib­il­ity of re­vers­ing time, it was im­port­ant to try and un­der­stand the re­la­tion­ship between the re­versed pro­cess and the dual pro­cess.

In or­der to pre­serve the tem­por­al ho­mo­gen­eity of the re­versed pro­cess, it was ne­ces­sary to re­verse time from a ran­dom time \( \zeta \), so that the re­versed pro­cess took the form \( \hat X(t) = X(\zeta-t) \) for \( 0 < t < \zeta \), where \( X \) is the ori­gin­al pro­cess and \( \zeta \) is a ran­dom time, sub­ject to cer­tain hy­po­theses that guar­an­teed that \( \hat X \) was tem­por­ally ho­mo­gen­eous. Ini­tially \( \zeta \) was taken to be the “life­time” of \( X \), but later this was ex­ten­ded to more gen­er­al times. Hunt con­sidered this prob­lem in the case of dis­crete-para­met­er Markov chains in 1960, and Chung con­sidered the case of a class of con­tinu­ous-para­met­er Markov chains in 1962. In the mid-six­ties, this work was ex­ten­ded to the class of Markov pro­cesses de­scribed above, in a series of pa­pers as­so­ci­ated with the names of Ike­da, Kunita, Na­gas­awa, Sato, S. Watanabe and T. Watanabe. However, these last-named au­thors as­sumed the ex­ist­ence of a pair of semig­roups (or re­solvents) in du­al­ity, the dual semig­roup be­ing the semig­roup of the pu­tat­ive dual or re­versed pro­cess. Moreover, in or­der to force the dual or re­versed pro­cess in­to the class of right-con­tinu­ous strong Markov pro­cesses, they re­placed \( \hat X(t) \) by its right lim­it; that is, the dual pro­cess was taken to be \( \tilde X(t) = \hat X(t+) \), so that the path \( t\mapsto \tilde X(t) \) is right con­tinu­ous.

A fun­da­ment­al break­through was made in the 1969 pa­per of Chung and Walsh, To re­verse a Markov pro­cess [1]. The au­thors did not try to fit the re­versal in­to the right-con­tinu­ous strong Markov frame­work as in most pre­vi­ous work. They worked dir­ectly with the left-con­tinu­ous pro­cess \( \hat X(t)= X(\zeta-t) \), as defined above. Much more im­port­antly, they did not as­sume the ex­ist­ence of a dual semig­roup; rather, part of their con­struc­tion of a good dual pro­cess was con­struct­ing a good dual semig­roup. Moreover, they real­ized that the strong Markov prop­erty had to be mod­i­fied when the pro­cess had left-con­tinu­ous paths. They defined a new prop­erty that they called the mod­er­ately strong Markov prop­erty, and showed that this new prop­erty was the prop­er re­place­ment of the strong Markov prop­erty when the paths were left con­tinu­ous. P. A. Mey­er wrote that this pa­per of Chung and Walsh “made very im­port­ant (très grand) pro­gress in the the­ory of re­vers­ing a Markov pro­cess.” Para­phras­ing slightly, he went on to say that the ori­gin­al­ity of this pa­per con­sisted pre­cisely in us­ing the nat­ur­al left-con­tinu­ous re­versal, and prov­ing that it sat­is­fied a “mod­er­ate” Markov prop­erty that is the nat­ur­al form of the strong Markov prop­erty for left-con­tinu­ous Markov pro­cesses. Of course, this pa­per con­tains much more than what I have men­tioned here, and I have only made su­per­fi­cial re­marks about those parts I have dis­cussed. In par­tic­u­lar, the con­struc­tion of a good dual semig­roup was tech­nic­ally dif­fi­cult, and in­tro­duced meth­ods that had not been used pre­vi­ously in the work on Markov pro­cesses; for ex­ample, the use of es­sen­tial lim­its.

The some­what awk­ward name “mod­er­ately strong Markov prop­erty” quickly evolved in­to the more suc­cinct “mod­er­ate Markov prop­erty.” This is not a weak form of the strong Markov prop­erty, but a dis­tinct prop­erty that is ap­pro­pri­ate for left-con­tinu­ous pro­cesses. (More lo­gic­al ter­min­o­logy might be “right Markov” and “left Markov” for “strong Markov” and “mod­er­ate Markov”, re­spect­ively.) The re­la­tion­ship between these prop­er­ties was cla­ri­fied by Chung in his 1972 pa­per, On the fun­da­ment­al hy­po­theses of Hunt pro­cesses [2]. Left-con­tinu­ous mod­er­ate Markov pro­cesses were stud­ied as pro­cesses in their own right in a joint pa­per by Chung and Glover in 1979, Left-con­tinu­ous mod­er­ate Markov pro­cesses [4]. Since the mid-eighties, the ex­ist­ence of a left-con­tinu­ous mod­er­ate Markov dual pro­cess has been used ex­tens­ively in the study of the po­ten­tial the­ory of Markov pro­cesses. There is an up­dated and much ex­pan­ded ex­pos­i­tion of the Chung–Walsh the­ory in the second half of their 2005 book Markov Pro­cesses, Browni­an Mo­tion and Time Sym­metry [7].

Re­gard­ing the use of last exit times, Chung had already made good use of such times in his work on Markov chains. In the work of Hunt, and for the next ten years fol­low­ing Hunt, a key concept was that of an op­tion­al time, of which the fun­da­ment­al ex­ample is a first entry (or hit­ting) time. Last exit times were hardly con­sidered, if at all, dur­ing this peri­od. But it is clear that a last exit time for \( X \) is just a first entry time for the re­versed pro­cess \( \hat X \) defined earli­er. Thus, over­sim­pli­fy­ing again, the study of last exit times may be re­duced to that of the more fa­mil­i­ar first entry times, but for the re­versed pro­cess.

In his pa­per Prob­ab­il­ist­ic ap­proach in po­ten­tial the­ory to the equi­lib­ri­um prob­lem [3], Chung made spec­tac­u­lar use of last exit times to ob­tain a beau­ti­ful for­mula for the equi­lib­ri­um dis­tri­bu­tion of a set, in terms of the last exit dis­tri­bu­tion from the set and the po­ten­tial ker­nel of the un­der­ly­ing pro­cess. Since I have com­men­ted on this pa­per in some de­tail in the volume of his se­lec­ted works, I will con­fine my­self here to this: This pa­per was very ori­gin­al in its meth­od, and this was im­me­di­ately re­cog­nized by work­ers in prob­ab­il­ist­ic po­ten­tial the­ory.

Of course, the above is just a small sample of the many im­port­ant con­tri­bu­tions that Kai Lai Chung made to the the­ory of Markov pro­cesses and prob­ab­il­ist­ic po­ten­tial the­ory.

Works

[1]K. L. Chung and J. B. Walsh: “To re­verse a Markov pro­cess,” Acta Math. 123 : 1 (1969), pp. 225–​251. MR 0258114 Zbl 0187.​41302 article

[2]K. L. Chung: “On the fun­da­ment­al hy­po­theses of Hunt pro­cesses,” pp. 43–​52 in Con­ve­gno di cal­colo delle prob­ab­il­ità (IN­DAM, Rome, March–April, 1971). Edi­ted by F. Severi. Sym­po­sia Math­em­at­ica IX. Aca­dem­ic Press (Lon­don), 1972. MR 0359019 Zbl 0242.​60031 incollection

[3]K. L. Chung: “Prob­ab­il­ist­ic ap­proach in po­ten­tial the­ory to the equi­lib­ri­um prob­lem,” Ann. Inst. Four­i­er (Gren­oble) 23 : 3 (1973), pp. 313–​322. MR 0391277 Zbl 0258.​31012 article

[4]K. L. Chung and J. Glover: “Left con­tinu­ous mod­er­ate Markov pro­cesses,” Z. Wahr­sch. Verw. Ge­bi­ete 49 : 3 (1979), pp. 237–​248. MR 547825 Zbl 0413.​60063 article

[5]K. L. Chung: Lec­tures from Markov pro­cesses to Browni­an mo­tion. Grundlehren der Math­em­at­ischen Wis­senschaften 249. Spring­er (New York), 1982. MR 648601 Zbl 0503.​60073 book

[6]K. L. Chung: Green, Brown, and prob­ab­il­ity. World Sci­entif­ic (River Edge, NJ), 1995. MR 1371379 Zbl 0871.​60001 book

[7]K. L. Chung and J. B. Walsh: Markov pro­cesses, Browni­an mo­tion, and time sym­metry, 2nd edition. Grundlehren der Math­em­at­ischen Wis­senschaften 249. Spring­er (New York), 2005. MR 2152573 Zbl 1082.​60001 book