Celebratio Mathematica

I first met Kai Lai Chung in 1955 when he gave a sem­in­ar talk at Prin­ceton, where I was an in­struct­or at the time. I be­lieve that he spoke about his work on Markov chains. After so many years I re­mem­ber very little about the talk, but I clearly re­mem­ber how im­pressed I was with the en­thu­si­asm and en­ergy dis­played by the speak­er. I had the pleas­ure of spend­ing the aca­dem­ic year 1964–65 at Stan­ford, and dur­ing that time Kai Lai and I be­came good friends. We had the op­por­tun­ity to dis­cuss math­em­at­ics in some depth, and we of­ten had lunch to­geth­er. He had be­come in­ter­ested in po­ten­tial the­ory, and had in­vited Mar­cel Brelot to vis­it Stan­ford dur­ing the spring quarter of 1965, and give a course on clas­sic­al po­ten­tial the­ory. By the end of the term, he and I were the only ones still at­tend­ing Brelot’s lec­tures! Dur­ing the 1970s we had an ex­tens­ive cor­res­pond­ence about Markov pro­cesses, prob­ab­il­ist­ic po­ten­tial the­ory and re­lated top­ics. In­ter­act­ing with Kai Lai on any level was al­ways ex­tremely stim­u­lat­ing and re­ward­ing.

In what fol­lows I am go­ing to to com­ment on some of his work that was es­pe­cially im­port­ant and in­flu­en­tial in areas that are of in­terest to me.

Dur­ing the early 1970s there was a con­sid­er­able body of work on what might be called the gen­er­al the­ory of ex­cur­sions of a Markov pro­cess. Per­haps the defin­it­ive work in this dir­ec­tion was the pa­per of Mais­on­neuve [e9]. Shortly there­after Chung’s pa­per [6] on Browni­an ex­cur­sions ap­peared. Some of his res­ults had been an­nounced earli­er in [5]. Chung did not make use of the gen­er­al the­ory; rather, work­ing by hand, he made a deep study of the ex­cur­sions of Browni­an mo­tion from the ori­gin, us­ing the spe­cial prop­er­ties of Browni­an mo­tion. This pa­per was a tour de force of dir­ect meth­ods for pen­et­rat­ing the mys­ter­ies of these ex­cur­sions. Guided some­what, it seems, by ana­logy with his pre­vi­ous work on Markov chains, and in­spired by Lévy’s work, he ob­tained a wealth of ex­pli­cit for­mu­las for the dis­tri­bu­tions of vari­ous ran­dom vari­ables and pro­cesses de­rived from an ex­cur­sion. I shall de­scribe briefly a few of his res­ults, without re­pro­du­cing the de­tailed ex­pli­cit ex­pres­sions in the pa­per.

Let $B=B(t)$ de­note one-di­men­sion­al Browni­an mo­tion start­ing from the ori­gin, and let $Y=|B|$. Fix $t>0$. Fol­low­ing Chung, define $$\begin{array}{rl}\gamma (t)& =sup\{s\le t:Y(s)=0\},\\ \beta (t)& =inf\{s\ge t:Y(s)=0\}.\end{array}$$ The in­ter­vals $(\gamma (t),\beta (t))$ and $(\gamma (t),t)$ are called the ex­cur­sion in­ter­val strad­dling $t$, and the in­ter­val of me­an­der­ing end­ing at $t$, re­spect­ively. Let $L(t)=\beta (t)-\gamma (t)$ and $L^{-}(t)=t-\gamma (t)$ de­note the lengths of these in­ter­vals. Chung be­gins by giv­ing a dir­ect de­riv­a­tion of a num­ber of res­ults, ori­gin­ally due to Lévy, which lead to the joint dis­tri­bu­tion of $(\gamma (t),Y(t),\beta (t))$. Moreover, based on his earli­er work on Markov chains, he is able to write these for­mu­las in a par­tic­u­larly il­lu­min­at­ing form. Define $$\begin{array}{rl}{Z}^{-}(u)& =Y(\gamma (t)+u)\text{for }0\le u\le L^{-}(t),\\ Z(u)& =Y(\gamma (t)+u)\text{for }0\le u\le L(t).\end{array}$$ Then $Z^{-}$ is called the me­an­der­ing pro­cess, and $Z$ the ex­cur­sion pro­cess. The­or­em 4 gives the joint law of $\gamma (t)$ and $Z^{-}$, while The­or­em 6 con­tains the joint law of $\gamma (t),L(t)$ and $Z$. (Chung de­notes both the me­ander pro­cess and the ex­cur­sion pro­cess by $Z$; I have changed the nota­tion for this ex­pos­i­tion.) Chung then ap­plies these res­ults to cal­cu­late the dis­tri­bu­tions of vari­ous func­tion­als of these pro­cesses. Par­tic­u­larly in­ter­est­ing is The­or­em 7, which con­tains an ex­pli­cit for­mula for the max­im­um of $Z$ con­di­tioned on $\gamma (t)$ and $L(t)$. A con­sequence is that $$F(x)=1+2\sum _{n=1}^{\mathrm{\infty }}(1-2nx)e^{-n^{2}x}\text{for }0<x<\mathrm{\infty }$$ defines a dis­tri­bu­tion func­tion! This is dis­cussed in some de­tail. Oth­er func­tion­als were also stud­ied. Of spe­cial in­terest to me is the oc­cu­pa­tion time of an in­ter­val $(a.b)$ dur­ing an ex­cur­sion defined by $$S(t,a.b)={\int }_{\gamma (t)}^{\beta (t)}{1}_{(a.b)}(Z(s))ds.$$ Among oth­er things, Chung showed that $S(t,0,\epsilon )/{\epsilon }^2$ has a lim­it­ing dis­tri­bu­tion as $\epsilon \downarrow 0$, and com­puted its first four mo­ments. In [e10] it was shown that this dis­tri­bu­tion was the con­vo­lu­tion of the first pas­sage dis­tri­bu­tion $P(R\in ds)$ with it­self, where $R=inf\{s:Y(s)=1\}$.

In the pa­per [7] some of the ba­sic prop­er­ties of a left-con­tinu­ous mod­er­ate Markov pro­cess were for­mu­lated and proved. It was more or less ig­nored when it first ap­peared, even though the im­port­ance of this class of pro­cesses was evid­ent from the fun­da­ment­al pa­per of Chung and Walsh [1] on time re­versal of Markov pro­cesses. In [1], it was called the mod­er­ately strong Markov prop­erty, and the pro­cess was right con­tinu­ous. To the best of my know­ledge, the ter­min­o­logy “mod­er­ate Markov prop­erty” first ap­peared in [2]. In 1987, Fitz­sim­mons [e11] was able to modi­fy some­what the Chung–Walsh meth­ods, and so to con­struct a left-con­tinu­ous mod­er­ate Markov dual pro­cess for any giv­en Borel right pro­cess and ex­cess­ive meas­ure $m$ as du­al­ity meas­ure. The Chung–Walsh the­ory cor­res­ponds to $m$ be­ing the po­ten­tial of a meas­ure $\mu$ which served as a fixed ini­tial dis­tri­bu­tion. More im­port­antly, Fitz­sim­mons showed that this dual was a power­ful tool in study­ing the po­ten­tial the­ory of the un­der­ly­ing Borel right pro­cess. Con­sequently, there was re­newed in­terest in left-con­tinu­ous mod­er­ate Markov pro­cesses, and the Chung–Glover pa­per [7] was im­me­di­ately rel­ev­ant. It has be­come the ba­sic ref­er­ence for prop­er­ties of these pro­cesses.

The pa­per [3] was per­haps Chung’s most in­flu­en­tial con­tri­bu­tion to what is com­monly known as prob­ab­il­ist­ic po­ten­tial the­ory. (This ex­cludes his work on gauge the­or­ems and Schrödinger equa­tions.) In it, he ob­tained a beau­ti­ful ex­pres­sion for the equi­lib­ri­um dis­tri­bu­tion of a set, in terms of the last-exit dis­tri­bu­tion from the set and the po­ten­tial ker­nel dens­ity of the un­der­ly­ing pro­cess. He em­phas­ized and clearly stated that his ap­proach in­volved work­ing dir­ectly with the last exit time. This was an im­port­ant in­nov­a­tion since such times are not stop­ping times, and so were not part of the avail­able ma­chinery at that time. Im­me­di­ately fol­low­ing Chung’s pa­per (more pre­cisely, its an­nounce­ment) and in­spired by it, Mey­er [e8] and Getoor and Sharpe [e7] ob­tained sim­il­ar res­ults un­der dif­fer­ent hy­po­theses. Nu­mer­ous au­thors then de­veloped tech­niques for hand­ling last exit and more gen­er­al times, which be­came part of the stand­ard ma­chinery of Markov pro­cesses. In two ad­di­tion­al pa­pers, [4] and (with K. Mur­ali Rao) [8], con­di­tions were giv­en un­der which the equi­lib­ri­um meas­ure ob­tained from the last-exit dis­tri­bu­tion is a mul­tiple of the meas­ure of min­im­um en­ergy, as in clas­sic­al situ­ations. In [8], sym­metry was not as­sumed, and so a mod­i­fied form of en­ergy was in­tro­duced in or­der to ob­tain reas­on­able res­ults. Ad­di­tion­al im­plic­a­tions in po­ten­tial the­ory of the hy­po­theses that he had in­tro­duced in [3], and also their re­la­tion­ship with the more com­mon du­al­ity hy­po­theses, were ex­plored with K. Mur­ali Rao in [9] and with Ming Liao and Rao in [10]. Of par­tic­u­lar im­port­ance was the res­ult giv­ing suf­fi­cient con­di­tions for the valid­ity of Hunt’s hy­po­thes­is B in [9]. These four pa­pers were very ori­gin­al, but for some reas­on they were not as in­flu­en­tial as the pa­per [3].

For his­tor­ic­al reas­ons, I should point out that the re­la­tion­ship between the equi­lib­ri­um meas­ure and the last-exit dis­tri­bu­tion had ap­peared a few years earli­er in Port and Stone’s mem­oir on in­fin­itely di­vis­ible pro­cesses — see sec­tions 8 and 11 of [e6]. One may won­der why Chung’s pa­per [3], was im­me­di­ately so in­flu­en­tial, while the res­ult in Port and Stone was hardly no­ticed at the time. Cer­tainly it was un­known to Chung, and evid­ently Mey­er was also un­aware of it. The most likely reas­ons for this are two-fold: (1) The res­ult in Port and Stone was bur­ied in a mem­oir of just over two hun­dred pages; in ad­di­tion, their proof of the in­teg­ral con­di­tion for the tran­si­ence or re­cur­rence of an in­fin­itely di­vis­ible pro­cess at­trac­ted the most at­ten­tion at the time. (2) In Chung it was the main res­ult of the pa­per, it was clearly stated, and proved by a dir­ect eas­ily un­der­stood ar­gu­ment.

I shall now ex­plain in a bit more de­tail what Chung did. I’ll try to em­phas­ize the ideas, leav­ing aside the tech­nic­al­it­ies. So, sup­pose that $X=(X_t,P^x)$ is a Hunt pro­cess tak­ing val­ues in a loc­ally com­pact, sep­ar­able Haus­dorff space $E$. If $B\in \mathcal{E}$ (the $\sigma$-al­gebra of Borel sub­sets of $E$), define the hit­ting time $T_B$ and the last exit time ${\lambda }_B$ of $B$ by $$T_B=inf\{t>0:X_t\in B\}\text{and}{\lambda }_B=sup\{t>0:X_t\in B\},$$ where the in­fim­um (re­spect­ively, su­prem­um) of the empty set is $\mathrm{\infty }$ (re­spect­ively, 0). Let $$U(x,B)=E^x{\int }_0^{\mathrm{\infty }}{1}_B(X_t)dt$$ de­note the po­ten­tial ker­nel of $X$, and sup­pose that $U(\cdot ,K)$ is bounded for $K$ com­pact; in par­tic­u­lar, $X$ is tran­si­ent. For the mo­ment, sup­pose $X$ is a Browni­an mo­tion in ${\mathbb{R}}^d$ for $d\ge 3$. Then, $$U(x,B)={\int }_{B}u(x,y)dy$$ where $u(x,y)=c_d|x-y{|}^{2-d}$ is the New­to­ni­an po­ten­tial ker­nel ap­pro­pri­ately nor­mal­ized. A clas­sic­al res­ult in po­ten­tial the­ory states that if $K\subset {\mathbb{R}}^d$ is com­pact and has pos­it­ive (New­to­ni­an) ca­pa­city, then there ex­ists a unique meas­ure ${\mu }_K$, called the equi­lib­ri­um meas­ure or dis­tri­bu­tion of $K$, car­ried by $K$ and whose po­ten­tial $$\begin{array}{}\text{(1)}& p_K(x)=U{\mu }_K(x)=\int u(x,y){\mu }_k(dy)\end{array}$$ is less than or equal to 1 every­where, and takes the value 1 on $K$. Ac­tu­ally, $p_K\equiv 1$ on $K$ only if $K$ is reg­u­lar; in gen­er­al, there may be an ex­cep­tion­al sub­set of $K$ of ca­pa­city zero on which $p_K<1$. The func­tion $p_K$ is called the equi­lib­ri­um po­ten­tial of $K$, and may be char­ac­ter­ized as the unique su­per­har­mon­ic func­tion $v$ on ${\mathbb{R}}^d$ such that $0\le v\le 1$, $v$ is har­mon­ic on ${\mathbb{R}}^d\mathrm{\setminus }K$, and $\{v<1\}\cap K$ has ca­pa­city zero — $v\equiv 1$ on $K$ if $K$ is reg­u­lar. Evid­ently Kak­utani [e1] was the first per­son to note that $$\begin{array}{}\text{(2)}& p_K(x)=P^x(T_K<\mathrm{\infty })=P^x(X_t\in K\text{ for some }t>0).\end{array}$$ One may ask for what class of Borel sets $B\subset {\mathbb{R}}^d$ does there ex­ist a meas­ure ${\mu }_B$ such that $$\begin{array}{}\text{(3)}& P^x(T_B<\mathrm{\infty })=\int u(x,y){\mu }_B(dy),\end{array}$$ and what can be said about ${\mu }_B$. This is the equi­lib­ri­um prob­lem, as stated in the first para­graph of Chung’s pa­per.

Now re­turn to the situ­ation in which $X$ is a Hunt pro­cess, as de­scribed in the first few sen­tences of the pre­ced­ing para­graph. For $B\in \mathcal{E}$, re­call the defin­i­tions of the hit­ting time $T_B$ and the last exit time ${\lambda }_B$. The set $B$ is tran­si­ent, provided $P^x({\lambda }_B<\mathrm{\infty })=1$ for all $x$. Also note that $$p_B(x)=P^x(T_B<\mathrm{\infty })=P^x({\lambda }_B>0).$$ Fix $B$ tran­si­ent, and let $p=p_B$. It is eas­ily checked that $p$ is ex­cess­ive, and $P_{t}p\to 0$ as $t\to \mathrm{\infty }$. Here, $P_t=(P_t(x,\cdot ))$ is the trans­ition semig­roup of $X$. Form­ally, from semig­roup the­ory, $(p-P_{\epsilon }p)/\epsilon \to -\mathcal{G}p$, where $\mathcal{G}$ is the “gen­er­at­or” of $(P_t)$, and $p=U(-\mathcal{G}p)$, with $U$ the po­ten­tial ker­nel of $X$ as defined above. Of course, in gen­er­al $p$ is not in the do­main of $\mathcal{G}$. However, if we want to rep­res­ent $p$ as the po­ten­tial of something, then one ex­pects it to be some sort of lim­it of $p_{\epsilon }=(p-P_{\epsilon }p)/\epsilon$ as $\epsilon \downarrow 0$. This idea had been used by McK­ean and Tana­ka [e4], Volkon­ski [e3] and Šur [e5] to rep­res­ent ex­cess­ive func­tions as po­ten­tials of ad­dit­ive func­tion­als. More rel­ev­ant to the present dis­cus­sion, us­ing the same ba­sic idea, Hunt [e2] had shown, for what are now called Hunt pro­cesses sat­is­fy­ing, in ad­di­tion, the ex­ist­ence of a nice dual pro­cess and sub­ject to a type of Feller con­di­tion and a tran­si­ence hy­po­thes­is, that, if $B$ has com­pact clos­ure, then $\text{(3)}$ holds, where now $u(x,y)$ is the po­ten­tial dens­ity as­so­ci­ated with $X$ and its dual; in par­tic­u­lar, $U(x,dy)=u(x,y)m(dy)$, where $m$ is the du­al­ity meas­ure — Le­besgue meas­ure when $X$ is Browni­an mo­tion.

Chung’s key ob­ser­va­tion was to note that $$p-P_{\epsilon }p=P^{\cdot }({\lambda }_B>0)-P^{\cdot }({\lambda }_B>\epsilon )=P^{\cdot }(0<{\lambda }_B\le \epsilon ).$$ Sup­pose $f\ge 0$ is a bounded con­tinu­ous func­tion, and for sim­pli­city write $\lambda ={\lambda }_B$. Then, by the Markov prop­erty, $$\begin{array}{rl}U[f(p-P_{\epsilon }p)]& =E^{\cdot }{\int }_0^{\mathrm{\infty }}f(X_t)P^{X(t)}(0<\lambda \le \epsilon )dt\\ & =E^{\cdot }{\int }_0^{\mathrm{\infty }}f(X_t)1_{\{0<\lambda \circ {\theta }_t\le \epsilon \}}dt.\end{array}$$ Here, ${\theta }_t$ is the shift op­er­at­or which shifts the ori­gin of the path from 0 to $t$ so that $X_s\circ {\theta }_t=X_{s+t}$ for $s\ge 0$. It is eas­ily checked that $\lambda \circ {\theta }_t=(\lambda -t{)}^{+}$. Plug­ging this in­to the last in­teg­ral and re­call­ing that $p_{\epsilon }=(p-P_{\epsilon }p)/\epsilon$, one finds $$\begin{array}{rl}\text{(4)}& U[f{p}_{\epsilon }]& =\frac{1}{\epsilon }{E}^{\cdot }[{\int }_{(\lambda -\epsilon {)}^{+}}^{\lambda }f(X_t)dt;\lambda >0]& \to E^x[f(X_{\lambda -}),\lambda >0]\text{as }\epsilon \downarrow 0.\end{array}$$ Sup­pose that there ex­ists a Radon meas­ure $m$ on $E$ such that $U(x,dy)=u(x,y)m(dy)$. Then, Chung im­posed ana­lyt­ic con­di­tions on the po­ten­tial dens­ity $u(x,y)$ which im­plied the ex­ist­ence of a meas­ure ${\mu }_B$ such that $$\begin{array}{rl}U[f{p}_{\epsilon }](x)& =\int u(x,y)f(y)p_{\epsilon }(y)m(dy)\\ & \to \int u(x,y)f(y){\mu }_B(dy)=U[f{\mu }_B](x)\text{as }\epsilon \downarrow 0\end{array}$$ for all bounded con­tinu­ous $f$ with com­pact sup­port. Com­bin­ing this with $\text{(4)}$, we ob­tain $$\begin{array}{}\text{(5)}& E^x[f(X_{\lambda -});\lambda >0]=U[f{\mu }_B](x),\end{array}$$ and tak­ing a se­quence of such $f$ in­creas­ing to 1, $$\begin{array}{}\text{(6)}& p_B(x)=P^x[T_B<\mathrm{\infty }]=P^x[{\lambda }_B>0]=U{\mu }_B(x).\end{array}$$ De­fin­ing the last-exit dis­tri­bu­tion $L_B(x,dy)=P^x[X_{\lambda -}\in dy,\lambda >0]$, $\text{(5)}$ im­plies that $$\begin{array}{}\text{(7)}& L_B(x,dy)=u(x,y){\mu }_B(dy).\end{array}$$ This for­mula $\text{(7)}$ is the cel­eb­rated res­ult of Chung which gives the prob­ab­il­ist­ic mean­ing of the equi­lib­ri­um meas­ure ${\mu }_B$. The meas­ure ${\mu }_B$ is car­ried by $\bar{B}$, even by $\partial B$ when $X$ has con­tinu­ous paths. Un­der Chung’s or Hunt’s hy­po­theses, ${\mu }_B$ is a Radon meas­ure; more gen­er­ally, un­der du­al­ity without Feller con­di­tions, ${\mu }_B$ is $\sigma$-fi­nite.

Let me de­rive a simple con­sequence of $\text{(5)}$, and for sim­pli­city I shall sup­pose $X$ is a Browni­an mo­tion in ${\mathbb{R}}^d$ with $d\ge 3$. Let $B\subset {\mathbb{R}}^d$ be tran­si­ent, for ex­ample with $\bar{B}$ com­pact. As be­fore, $\lambda ={\lambda }_B$. Since the paths are con­tinu­ous, $\text{(5)}$ and the Markov prop­erty im­ply that $$E^x[f(X_{\lambda });0<\lambda \le t]=Uf{\mu }_B(x)-P_{t}Uf{\mu }_B(x)$$ for $t>0$ and $f$ bounded with com­pact sup­port. Now, $P_t(x,dy)=g_t(y-x)dy$, where $g_t$ is the fa­mil­i­ar Gauss ker­nel. Hence, $$E^x[f(X_{\lambda });0<\lambda \le t]={\iint }_0^{t}ds{g}_s(y-x)f(y){\mu }_B(dy).$$ In­teg­rat­ing over ${\mathbb{R}}^d$ we ob­tain, since $g_s$ is a prob­ab­il­ity dens­ity, $${\int }_{{\mathbb{R}}^d}dx{E}^x[f(X_{\lambda });0<\lambda \le t]=t\int fd{\mu }_B;$$ that is, $$\begin{array}{}\text{(8)}& P^m[X_{\lambda }\in dy,\lambda \in dt]=dt{\mu }_B(dy)\text{for }t>0,\end{array}$$ where $m$ is Le­besgue meas­ure. Thus, $X_{\lambda }$ and $\lambda$ are in­de­pend­ent un­der the $\sigma$-fi­nite meas­ure $P^m$, and their joint dis­tri­bu­tion un­der $P^m$ is the product of ${\mu }_B$ and Le­besgue meas­ure. To my mind, this is one of the nicest prob­ab­il­ist­ic in­ter­pret­a­tions of the equi­lib­ri­um meas­ure for Browni­an mo­tion. Ac­tu­ally, this is val­id in much more gen­er­al­ity. For ex­ample, if $X$ has a strong dual and the du­al­ity meas­ure $m$ is in­vari­ant, then $$\begin{array}{}\text{(9)}& P^m(X_{\lambda -}\in dy,\lambda \in dt)=dt{\mu }_B(dy)\text{for }t>0.\end{array}$$ See [e7]. In par­tic­u­lar this holds for tran­si­ent Lévy pro­cesses in ${\mathbb{R}}^d$ whose po­ten­tial ker­nel is ab­so­lutely con­tinu­ous. In gen­er­al, if $m$ is not in­vari­ant, then $X_{\lambda }$ and $\lambda$ are not in­de­pend­ent un­der $P^m$.