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{align*} \gamma(t) &=\sup\{s\le t : Y(s) =0\},\\ \beta(t) &= \inf\{s\ge t: Y (s) = 0\}. \end{align*} 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{align*} Z^-(u) &= Y(\gamma(t) + u) \quad\text{for } 0 \le u \le L^-(t),\\ Z(u) &= Y(\gamma(t) +u) \quad\text{for }0\le u \le L(t). \end{align*} 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 \[ F(x) = 1 +2 \sum^\infty_{n=1} (1-2nx) e^{-n^2x} \quad\text{for } 0 < x < \infty \] 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 \[ S(t, a{.}b) = \int^{\beta(t)}_{\gamma(t)} 1_{(a.b)} (Z(s))\,ds. \] Among oth­er things, Chung showed that \( S(t, 0,\varepsilon)/\varepsilon^2 \) has a lim­it­ing dis­tri­bu­tion as \( \varepsilon \downarrow 0 \), 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 \[ T_B= \inf\{t > 0: X_t \in B\} \quad\text{and}\quad \lambda_B = \sup \{t > 0: X_t \in B\}, \] where the in­fim­um (re­spect­ively, su­prem­um) of the empty set is \( \infty \) (re­spect­ively, 0). Let \[ U(x, B) = E^x \int^\infty_0 1_B(X_t) \,dt \] 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, \[ U(x, B) = \int_Bu(x, y) \,dy \] 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{equation}\label{1} p_K(x) = U\mu_K(x) = \int u(x, y)\,\mu_k(dy) \end{equation} 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 \( \mathbb{R}^d\backslash K \), 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{equation}\label{2} p_K(x) = P^x(T_K < \infty) = P^x(X_t \in K \text{ for some } t > 0). \end{equation} One may ask for what class of Borel sets \( B\subset \mathbb{R}^d \) does there ex­ist a meas­ure \( \mu_B \) such that \begin{equation}\label{3} P^x(T_B < \infty) = \int u(x, y) \,\mu_B(dy), \end{equation} 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 \( P^x(\lambda_B < \infty)=1 \) for all \( x \). Also note that \[ p_B(x) = P^x(T_B < \infty) = P^x(\lambda_B > 0). \] Fix \( B \) tran­si­ent, and let \( p = p_B \). It is eas­ily checked that \( p \) is ex­cess­ive, and \( P_t p \to 0 \) as \( t\to \infty \). Here, \( P_{t} = (P_t(x,\,\cdot\,)) \) is the trans­ition semig­roup of \( X \). Form­ally, from semig­roup the­ory, \( (p-P_\varepsilon p)/\varepsilon \to - \mathcal{G}p \), 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 \( p_\varepsilon = (p - P_{\varepsilon} p)/\varepsilon \) as \( \varepsilon \downarrow 0 \). 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 \eqref{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 \[ p-P_{\varepsilon}p = P^{\centerdot}(\lambda_B > 0) - P^{\centerdot}(\lambda_B > \varepsilon) = P^{\centerdot}(0 < \lambda_B \le \varepsilon). \] Sup­pose \( f\ge 0 \) is a bounded con­tinu­ous func­tion, and for sim­pli­city write \( \lambda = \lambda_B \). Then, by the Markov prop­erty, \begin{align*} U\bigl[f(p -P_{\varepsilon}p)\bigr] &= E^{\centerdot} \int^\infty_0 f(X_t) \,P^{X(t)}(0 < \lambda \le\varepsilon) \,dt \\ &= E^{\centerdot} \int^\infty_0 f(X_t) \,1_{\{0 < \lambda \circ \theta_t \le \varepsilon\}}\, dt. \end{align*} 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 \( p_{\varepsilon}=(p -P_{\varepsilon}p)/\varepsilon \), one finds \begin{align} \label{4} U[fp_{\varepsilon}] &= \frac 1{\varepsilon} E^{\centerdot} \Bigl[\int^\lambda_{(\lambda - {\varepsilon})^+} f(X_t) \,dt; \lambda > 0\Bigr] \\ &\to E^x\bigl[f(X_{\lambda-}), \lambda > 0\bigr] \qquad\text{as }\varepsilon\downarrow 0. \nonumber \end{align} 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{align*} U[fp_{\varepsilon}](x) &= \int u(x, y) f(y) p_{\varepsilon}(y) \,m(dy) \\ &\to \int u(x, y) f(y) \mu_B (dy) = U[f\mu_B] (x) \qquad\text{as }\varepsilon\downarrow 0 \end{align*} for all bounded con­tinu­ous \( f \) with com­pact sup­port. Com­bin­ing this with \eqref{4}, we ob­tain \begin{equation}\label{5} E^x[f(X_{\lambda -}); \lambda > 0 ] = U[f\mu_B] (x), \end{equation} and tak­ing a se­quence of such \( f \) in­creas­ing to 1, \begin{equation}\label{6} p_B(x) = P^x [ T_B < \infty] = P^x[\lambda_B > 0] =U\mu_B(x). \end{equation} De­fin­ing the last-exit dis­tri­bu­tion \( L_B(x, dy) = P^x[X_{\lambda-} \in dy, \lambda > 0] \), \eqref{5} im­plies that \begin{equation}\label{7} L_B(x, dy) = u(x, y) \,\mu_B(dy). \end{equation} This for­mula \eqref{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 \( \overline B \), 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 \eqref{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 \( \overline B \) com­pact. As be­fore, \( \lambda = \lambda_B \). Since the paths are con­tinu­ous, \eqref{5} and the Markov prop­erty im­ply that \[ E^x\bigl[f(X_\lambda); 0 < \lambda \le t \bigr] = Uf\mu_B(x) - P_t Uf \mu_B(x) \] 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, \[ E^x\bigl[f(X_\lambda); 0 < \lambda \le t\bigr] = \iint^t_0 ds\ g_s (y-x) f(y)\,\mu_B(dy). \] In­teg­rat­ing over \( \mathbb{R}^d \) we ob­tain, since \( g_s \) is a prob­ab­il­ity dens­ity, \[ \int_{\mathbb{R}^d} dx\, E^x\bigl[f(X_\lambda); 0 < \lambda \le t\bigr] = t \int f\,d\mu_B; \] that is, \begin{equation}\label{8} P^m \bigl[X_\lambda \in dy, \lambda \in dt\bigr] = dt \,\mu_B(dy) \quad\text{for }t > 0, \end{equation} 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{equation}\label{9} P^m\bigl(X_{\lambda-} \in dy, \,\lambda \in dt\bigr) = dt\,\mu_B (dy) \quad\text{for }t > 0. \end{equation} 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 \( P^m \).