#### by Ronald Getoor

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.

#### Excursions

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 __\( R=\inf \{s: Y(s) =1\} \)__.

#### Moderate Markov processes

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.

#### Probabilistic potential theory

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 __\( P^m \)__.