#### by Ruth Williams

In the late 1970s, Kai Lai Chung began investigating connections between probability
and the (reduced) Schrödinger equation:
__\begin{equation}\label{RW-reds}
\tfrac{1}{2} \Delta u(x) + q(x)u(x) = 0 \quad \hbox{for } x\in \mathbb{R}^d,
\end{equation}__
where __\( q \)__ is a real-valued Borel-measurable function on __\( \mathbb{R}^d \)__, and
__\( \Delta \)__ is the __\( d \)__-dimensional Laplacian.
His work in this area extended over the next 15 years or so. It included
collaborations with several colleagues and students, and inspired the work of
others. His book,
*From Brownian Motion to Schrödinger’s Equation*
[6],
written with Zhongxin Zhao, is a compilation and
refinement of much of the research conducted in this area up through 1994.

In the following, I will describe some of the background and early advances in this research involving connections with Brownian motion. A complementary article written by Michael Cranston, which also appears in this volume, focuses on related developments involving connections with conditioned Brownian motion. My account is not meant to be exhaustive, but rather to provide a sample of some of the intriguing aspects of the topic and to illustrate the pivotal role that Kai Lai Chung played in some of the developments. My description is necessarily influenced by my own personal recollections.

#### Background

Stimulated by Feynman’s
[e1]
proposed “path integral”
solution of the complex time-dependent
Schrödinger
equation, Kac
[e2],
[e3]
considered,
for a Borel-measurable function __\( q:\mathbb{R}\to \mathbb{R} \)__ satisfying __\( q\leq 0 \)__,
the following
multiplicative functional of one-dimensional Brownian motion __\( B \)__:
__\[ e_q(t) = \exp\Bigl( \int_0^t q(B_s)\, ds \Bigr)
\quad\text{for } t\geq 0, \]__
This functional can also be defined for suitable
Borel-measurable functions __\( q: \mathbb{R}^d
\to \mathbb{R} \)__, and for __\( B \)__ a __\( d \)__-dimensional
Brownian motion or even a __\( d \)__-dimensional
diffusion process. Such functionals are now called
*Feynman–Kac functionals*.

Consider a continuous bounded function __\( q:\mathbb{R}^d\to \mathbb{R} \)__,
and a continuous bounded function __\( g:\mathbb{R}^d\to\mathbb{R} \)__.
If __\( \psi:[0,\infty)\times \mathbb{R}^d\to \mathbb{R} \)__ is
a continuous bounded function, with continuous partial derivatives
__\( \partial \psi/\partial t \)__, __\( \partial \psi/\partial x_i \)__, and
__\( \partial^2 \psi/\partial x_i\partial x_j \)__
for __\( i,j=1,\ldots, d \)__,
defined on __\( (0,\infty)\times \mathbb{R}^d \)__,
and such that
the following time-dependent Schrödinger equation holds:
__\begin{equation} \label{RW-teds}
\frac{\partial \psi (t,x)}{\partial t} = \frac {1}{2} \Delta \psi(t,x) + q(x)
\psi(t, x)
\quad\text{for } t > 0, x\in \mathbb{R}^d,
\end{equation}__
with initial condition __\( \psi(0,x) = g(x) \)__ for __\( x\in\mathbb{R}^d \)__, and
where
__\begin{equation}
\Delta \psi (t,x) =\sum_{i=1}^d \frac{\partial^2 \psi}{\partial x_i^2} (t,x),
\end{equation}__
then
it can be shown (for example, by using
Itô’s formula),
that
__\begin{equation} \label{RW-expk}
\psi(t, x) = E^x \bigl[ e_q(t) g(B_t)\bigr]
\quad\text{for } t\geq 0, x\in \mathbb{R}^d.
\end{equation}__
Here, __\( E^x \)__ denotes the
expectation operator under which __\( B \)__ is a __\( d \)__-dimensional
Brownian motion starting from __\( x \)__.

Kac
[e2],
[e3]
was interested in __\eqref{RW-teds}__ when __\( d=1 \)__ and
__\( q\leq0 \)__.
However, rather than
considering this equation directly, he
worked with a reduced Schrödinger equation similar to __\eqref{RW-reds}__ (with
__\( q-s \)__ in place of __\( q \)__, and __\( d=1 \)__)
obtained by formally taking the Laplace transform (with parameter __\( s \)__)
in equation
__\eqref{RW-teds}__ to eliminate the time variable __\( t \)__.
Under mild conditions, for example when __\( q \)__ is bounded and continuous
in addition to being non-positive,
Kac
[e3]
showed that the Laplace transform of the right member of
__\eqref{RW-expk}__ with __\( g=1 \)__ satisfies this reduced Schrödinger equation.

In the late 1950s and early 1960s,
in developing a potential theory for Markov processes,
Dynkin (see
[e5],
Chapter XIII, §4, Theorem 13.16)
and others,
considered expressions of the form
__\begin{equation} \label{RW-fkX}
E^x\Bigl[\exp\Bigl(\int_0^{\tau} q(X_s)\,ds\Bigr) f(X_{\tau})\Bigr]
\quad\text{for } x\in \overline D,
\end{equation}__
where __\( X \)__ is a diffusion process in __\( \mathbb{R}^d \)__,
__\( \tau=\inf\{ s > 0: X_s\notin D\} \)__ is the first exit time of __\( X \)__ from
a bounded domain __\( D \)__ in __\( \mathbb{R}^d \)__,
__\( \overline D \)__ is the closure of __\( D \)__,
__\( f \)__ is a continuous function defined on the boundary of __\( D \)__,
and __\( q\leq0 \)__ is Hölder-continuous and bounded on __\( D \)__.
Here, __\( P^x \)__ and __\( E^x \)__ denote respectively probability and expectation operators
for __\( X \)__ starting from __\( x\in \overline D \)__.
The domain __\( D \)__ is *regular* if
__\[ P^x (\tau =0 ) =1 \quad \hbox{for each } x\in \partial D. \]__
Under suitable
assumptions on __\( X \)__ and assuming that __\( D \)__ is regular,
Dynkin showed that expressions of the form __\eqref{RW-fkX}__
yield continuous functions on __\( \overline D \)__ that satisfy
the equation
__\begin{equation}\label{RW-diffreds}
{\mathcal L} u(x) + q(x) u(x) = 0 \quad \hbox{for } x\in D,
\end{equation}__
with continuous boundary values given by __\( f \)__,
where __\( \mathcal L \)__ is the
infinitesimal generator of
the diffusion process __\( X \)__.
The assumption that __\( q\leq 0 \)__ implies that the action of the
stopped Feynman–Kac functional,
__\begin{equation} \label{RW-FKS}
e_q(\tau) = \exp\Bigl( \int_0^{\tau} q(X_s)\,ds\Bigr),
\end{equation}__
is to “kill” the diffusion process at a
state dependent exponential
rate given by __\( -q \)__ up until the stopping time __\( \tau \)__.
The assumption that __\( q \)__ is non-positive ensures
that the mean value of the stopped Feynman–Kac functional is always
finite; in fact, it is bounded by one.
In contrast,
Khasminskii
[e4]
considered
the case where __\( q \)__ is *non-negative*. In this case,
the action of the stopped Feynman–Kac functional can be interpreted as
“creating mass” at a state-dependent exponential rate given by __\( q \)__ up until
the stopping time __\( \tau \)__.
Accordingly, the expression in __\eqref{RW-fkX}__ can fail to
be finite if the domain is sufficiently large.
Indeed, the results of Khasminskii
[e4]
imply that (under similar conditions to those imposed by
Dynkin except that __\( q\geq 0 \)__)
the expression in __\eqref{RW-fkX}__ is finite for all __\( x\in \overline D \)__ if and only
if there is a continuous function __\( u \)__ that is strictly positive on __\( \overline D \)__
and satisfies
__\eqref{RW-diffreds}__.

It was not until the work of Chung
[2]
that probabilistic
solutions of __\eqref{RW-reds}__ for general (signed) __\( q \)__
became an object of considerable interest. The question of how
the oscillations of such a __\( q \)__ affect
the behavior of the stopped Feynman–Kac functional
__\eqref{RW-FKS}__ is an
intriguing one; in particular,
killing of mass in some locations may cancel creation
of mass in others. The next section
describes some of K. L. Chung’s
investigations on stopped Feynman–Kac functionals
with general __\( q \)__.

#### Feynman–Kac gauge and positive solutions of the Schrödinger equation

##### One-dimensional diffusions

Kai Lai Chung initiated his research on stopped Feynman–Kac functionals in
[2]
by considering a
one-dimensional diffusion process (i.e., continuous strong Markov process) __\( X \)__, and
the functional
__\eqref{RW-FKS}__ with bounded Borel-measurable __\( q:\mathbb{R}\to \mathbb{R} \)__, and
__\[ \tau=\tau_b\equiv \inf \{ t > 0: X_t =b\}
\quad\text{for }b\in \mathbb{R} .\]__
Assume that __\( P^x(\tau_b < \infty ) =1 \)__ for each __\( x\in \mathbb{R} \)__ and
__\( b\in \mathbb{R} \)__, and define
__\[
v(x,b) = E^x\Bigl[\exp\Bigl(\int_0^{\tau_b} q(X_s) \, ds \Bigr) \Bigr]
\quad\text{for } x\in \mathbb{R}, b\in \mathbb{R}.
\]__
Two fundamental properties of __\( v \)__ are that __\( 0 < v(x,b) \leq \infty \)__ for all
__\( x, b \in \mathbb{R} \)__, and that
__\[
v(a,b ) v(b,c) = v(a, c)
\]__
for any __\( a < b < c \)__ or __\( c < b < a \)__ in __\( \mathbb{R} \)__.
The latter follows from the strong Markov property. These properties lead
to the following:

For fixed __\( b\in \mathbb{R} \)__,

if

__\( v(x, b) < \infty \)__for some__\( x < b \)__, then__\( v(x, b) < \infty \)__for all__\( x < b \)__,if

__\( v(x, b) < \infty \)__for some__\( x > b \)__, then__\( v(x,b) < \infty \)__for all__\( x > b \)__.

Chung
[2]
introduced the following measures of finiteness of __\( v \)__:
__\begin{align*}
\alpha &= \inf\{ b\in \mathbb{R}: v(x, b) < \infty \ \hbox{ for all } x > b \}, \\
\beta&= \sup \{ b\in \mathbb{R}: v(x,b) < \infty\ \hbox{ for all } x < b\},
\end{align*}__
and
showed the next two results.

__\begin{align*}
\alpha &= \sup\{ b\in \mathbb{R}: v(x, b) =\infty \ \hbox{ for all } x > b \},\\
\beta &= \inf \{ b\in \mathbb{R}: v(x, b) =\infty \ \hbox{ for all } x < b \}.
\end{align*}__
Furthermore, if __\( \beta \in \mathbb{R} \)__, then
__\( v(x, \beta ) =\infty \)__ for all __\( x < \beta \)__, and,
if __\( \alpha \in \mathbb{R} \)__, then __\( v(x, \alpha ) = \infty \)__
for all __\( x > \alpha \)__.

The following conditions __(a)–(c)__ are equivalent.

__\( \beta =+\infty \)__.__\( \alpha =-\infty \)__.For all

__\( a, b\in \mathbb{R} \)__,__\( v(a,b)\,v(b,a) \leq 1 \)__.

Following on from this, Chung and Varadhan
[1]
proved the next two theorems for __\( q:\mathbb{R}\to \mathbb{R} \)__ bounded and continuous,
and __\( X \)__ a one-dimensional Brownian motion.

Fix __\( b\in \mathbb{R} \)__.
Suppose that __\( v(x,b) < \infty \)__ for some (and hence all) __\( x < b \)__. Then,
__\( u(x)=v(x , b) \)__ is twice continuously differentiable
for __\( x\in (-\infty , b) \)__, continuous for __\( x\in(-\infty, b] \)__, and
__\( u \)__
satisfies the reduced Schrödinger equation:
__\begin{equation}\label{RW-schrod}
\tfrac{1}{2} u^{\prime\prime}(x) + q u(x) =0
\quad\text{for } x \in (-\infty , b),
\end{equation}__
with the boundary condition __\( u(b) = 1 \)__.

The following conditions are equivalent.

There is a twice continuously differentiable, strictly positive function

__\( u \)__satisfying__\eqref{RW-schrod}__with__\( b=+\infty \)__.__\( \beta =+\infty \)__.__\( \alpha =-\infty \)__.For all

__\( a, b\in \mathbb{R} \)__,__\begin{equation} \label{RW-uab} v(a,b)\,v(b,a) \leq 1. \end{equation}__There is some pair of distinct real numbers

__\( a \)__,__\( b \)__such that__\eqref{RW-uab}__holds.

The equivalence of (a), (b) and (c) is an
analogue of Khasminskii’s
[e4]
results, but
with __\( d=1 \)__, __\( D=(-\infty, \infty) \)__ and __\( q \)__ allowed to change
sign.
Further discussion of this one-dimensional case for Brownian motion can be found in
Chapter 9 of the book by Chung and Zhao
[6].

##### Multidimensional Brownian motion

Chung soon moved on to consider multidimensional Brownian motions and domains of finite Lebesgue measure in the work [3] with K. Murali Rao. This paper appeared in the proceedings of the first “Seminar on Stochastic Processes”, held at Northwestern University in 1981. This series of annual conferences was initiated by K. L. Chung, E. Çinlar and R. K. Getoor. The “Seminars” have grown in size over the years, but the novel format of a few invited talks, with ample time reserved for less formal presentations and discussions, has persisted and is one of the attractions of these annual meetings held over two and a half days.

The paper [3] was a significant advance. In particular, it contained the first “gauge theorem.” It is stated in its original form below, and then some generalizations are mentioned.

For this, assume that
__\( B \)__ is a __\( d \)__-dimensional Brownian motion (__\( d\geq 1 \)__),
__\( P^x \)__ and __\( E^x \)__ are respectively probability and expectation operators
for __\( B \)__ starting from __\( x\in \mathbb{R}^d \)__,
__\( q:\mathbb{R}^d \to \mathbb{R} \)__ is a bounded Borel-measurable function,
__\( D \)__ is a domain in __\( \mathbb{R}^d \)__ with closure __\( \overline D \)__ and boundary
__\( \partial D \)__,
__\( m \)__ is __\( d \)__-dimensional Lebesgue measure, and
__\( f:\partial D\to \mathbb{R} \)__ is
a bounded Borel-measurable function with __\( f\geq 0 \)__.
Let
__\[
\tau^{}_D =\inf\{ t > 0: B_t \notin D\},
\]__
be the first exit time of __\( B \)__ from __\( D \)__.
Define
__\begin{equation} \label{RJW-defU}
u(D, q, f; x) = E^x \Bigl[ \exp\Bigl( \int_0^{\tau^{}_D } q(B_s)\,ds\Bigr) f(B_{\tau^{}_D}) ; \tau^{}_D < \infty\Bigr]
\quad\text{for } x\in \overline D.
\end{equation}__
When __\( m(D) < \infty \)__, __\( P^x(\tau^{}_D < \infty) =1 \)__ for all __\( x\in\overline D \)__
(see
[6],
Theorem 1.17)
and the qualifier __\( \tau^{}_D < \infty \)__
in __\eqref{RJW-defU}__ may be omitted.
The following is Theorem 1.2 in Chung and Rao
[3].

Suppose the domain __\( D \)__ satisfies __\( m(D) < \infty \)__. If
__\( u(D, q, f;\,\cdot\,) \not\equiv \infty \)__ in __\( D \)__, then
it is bounded in __\( \overline D \)__.

While visiting Chung at Stanford University in the early 1980s,
Zhongxin Zhao
[e7]
(see also
[6],
Theorems 5.19 and 5.20)
extended this result by relaxing
the assumptions on __\( q \)__ and __\( D \)__.
In particular, he showed that
the conclusion of
Theorem 6
continues to hold
if the boundedness condition on __\( q \)__
is relaxed to simply require that
__\( q:\mathbb{R}^d \to \overline{\mathbb{R}} \)__ is a Borel-measurable function satisfying
__\begin{equation} \label{RJW-Kato}
\lim_{\alpha \downarrow 0} \Bigl[ \sup_{x\in \mathbb{R}^d} \int_{|y-x|\leq \alpha}
\bigl|G(x-y) q(y) \bigr| \,dy \Bigr ] =0,
\end{equation}__
where __\( \overline{\mathbb{R}}=[-\infty, \infty] \)__ is the extended real line and,
for __\( x\in\mathbb{R}^d \)__,
__\[ G (x) =
\begin{cases}
|x|^{2-d} & \hbox{if } d\geq 3,\\
\ln \dfrac{1}{|x|} & \hbox{if } d=2, \\
|x| & \hbox{if } d=1. \\
\end{cases} \]__
The set of Borel-measurable functions __\( q:\mathbb{R}^d\to\overline{\mathbb{R}} \)__
satisfying __\eqref{RJW-Kato}__ is called the *Kato class*
(on __\( \mathbb{R}^d \)__), and is usually denoted by __\( K_d \)__ or __\( J \)__.
Various properties of these functions, as well as analytic properties of
associated weak solutions of the
reduced Schrödinger equation __\eqref{RJW-Sch}__, are described in
an extensive paper of Aizenman and Simon
[e6]
which appeared shortly after the work
[3]
of Chung and Rao.
The paper
[e6]
also describes connections between the stopped Feynman–Kac
functional
and weak solutions of __\eqref{RJW-Sch}__ under a spectral condition (see
Theorem 9
below).

Neil Falkner
also visited Chung at Stanford in the
early 1980s. During this time, Falkner
[e8]
proved a gauge theorem,
when conditioned Brownian motion is used in place of Brownian motion,
for bounded Borel-measurable __\( q \)__ and sufficiently smooth bounded domains __\( D \)__.
(Zhao subsequently used conditioned Brownian motion in his work
[e7].)
For more details on the work in
[e8]
and a discussion of subsequent generalizations,
see the article by Michael Cranston in this volume, and Chapter 7 of
the book by Chung and Zhao
[6].

The function __\( u(D, q, 1;\,\cdot\,) \)__ obtained by setting __\( f\equiv 1 \)__
is called the *gauge* (function) for __\( (D, q) \)__,
and we say that __\( (D,q) \)__ is *gaugeable* if
this function is bounded on __\( D \)__.

Under the assumptions of
Theorem 6
and assuming __\( (D,q) \)__ is gaugeable,
a second key result in the paper of Chung and Rao
[3]
provides sufficient conditions for __\( u(D, q, f;\,\cdot\,) \)__ to be a
twice continuously differentiable solution of
the reduced Schrödinger equation in __\( D \)__,
__\begin{equation} \label{RJW-Sch}
\tfrac{1}{2} \Delta u(x) + q(x) u(x) = 0 \quad \hbox{for } x\in D,
\end{equation}__
with continuous boundary values
given by __\( f \)__. As is usual in the theory of elliptic
partial differential equations, to ensure two
continuous derivatives for __\( u \)__, in dimensions two and higher,
one imposes a stronger condition on __\( q \)__ than simple continuity.
For example, locally Hölder-continuous functions are
often used.
For __\( d=1 \)__, let
__\( {\mathcal C}_1(D) \)__ denote the set of bounded continuous functions
__\( h: D \to \mathbb{R} \)__ and, for __\( d\geq 2 \)__, let __\( {\mathcal C}_d(D) \)__ denote the set of
bounded continuous functions __\( h: D\to \mathbb{R} \)__ such that, for each compact
set __\( K\subset D \)__, there are strictly positive, finite constants __\( \alpha, M \)__ such that
__\[
|h(x) -h(y) | \leq M |x-y|^\alpha
\quad\text{for all } x, y \in K.
\]__
The following theorem is proved in Chung and Rao
[3]
for __\( d\geq2 \)__. For __\( d=1 \)__, they impose local Hölder continuity on
__\( q \)__ to obtain the result, but
this condition can be relaxed by invoking a
suitable analytic lemma for a Green potential, as was shown in
Chung’s book
*Lectures from Markov Processes to Brownian Motion*
[4]
(see Proposition 4 of Section 4.7).
Note for this that,
for __\( d=1 \)__, __\( m(D) < \infty \)__ implies
that __\( D \)__ is a bounded interval.

Let __\( D \)__ be a regular domain in __\( \mathbb{R}^d \)__ satisfying __\( m(D) < \infty \)__.
Suppose that __\( q\in \mathcal C_d(D) \)__ and __\( f:\partial D\to\mathbb{R} \)__ is bounded
and continuous. Assume that __\( (D, q) \)__ is gaugeable, that is,
__\( u(D, q, 1; \,\cdot\,) \not=\infty \)__ in __\( D \)__. Then,
__\( u=u(D, q, f;\,\cdot\,) \)__ defined by __\eqref{RJW-defU}__ on __\( \overline D \)__ is twice continuously
differentiable in __\( D \)__, continuous and bounded on __\( \overline D \)__,
satisfies __\eqref{RJW-Sch}__ in __\( D \)__, and has
__\( u=f \)__ on __\( \partial D \)__.
Furthermore, __\( u \)__ is the unique twice continuously differentiable
solution of __\eqref{RJW-Sch}__
that is continuous and bounded on __\( \overline D \)__ and
agrees with __\( f \)__ on the boundary __\( \partial D \)__.

This theorem has been generalized to situations where __\( q \)__ is a
Kato-class function, and __\eqref{RJW-Sch}__
is interpreted in the weak sense of partial differential equation theory
(see
[6],
Section 4.4)

Note that under the assumptions of the theorem
above, if __\( f\geq 0 \)__, then __\( u(D, q, f;\,\cdot\,) \)__ is a non-negative
solution of __\eqref{RJW-Sch}__ and, if
__\( f > 0 \)__ on __\( \partial D \)__, then __\( u(D, q, f ;\,\cdot\,) > 0 \)__
on __\( \overline D \)__. One naturally expects there to be some relation
between the existence of such positive solutions of __\eqref{RJW-Sch}__ and
the sign of
__\[
\lambda (D,q) = \sup_\varphi \Bigl[ \int_D \bigl\{ -\tfrac{1}{2}|\nabla \varphi(x) |^2 + q(x) \varphi(x)^2\bigr\}\,dx \Bigr] ,
\]__
where the supremum is over all __\( \varphi :D \to \mathbb{R} \)__ such that
__\( \varphi \)__ is infinitely continuously differentiable in __\( D \)__,
has compact support in __\( D \)__, and satisfies
__\( \int_D\varphi(x)^2 dx =1 \)__.
The quantity __\( \lambda (D, q) \)__ is the supremum of the spectrum of
the operator __\( \frac{1}{2} \Delta + q \)__ on __\( L^2(D) \)__ (see
[6],
Proposition 3.29).
Indeed, there is a sharp relationship provided by the following theorem
(see Theorem 4.19 of
[6]
for a proof).

Let __\( D \)__ be a domain in __\( \mathbb{R}^d \)__ satisfying __\( m(D) < \infty \)__
and __\( q \)__ be a Kato-class function. Then, __\( (D, q) \)__ is gaugeable if and only if
__\( \lambda(D, q) < 0 \)__.

For bounded domains __\( D \)__,
Aizenman and Simon proved
in Theorem A.4.1 of
[e6]
the
“if” part of this theorem when __\( \lambda (D, q) < 0 \)__,
__\( u(D, q, f;\,\cdot\,) \)__ is a weak solution of __\eqref{RJW-Sch}__,
its boundary values are given by __\( f \)__, and they are assumed continuous if __\( f \)__ is
continuous and __\( D \)__ is regular.

The work of Chung and Rao
[3]
was the seed for
much subsequent work on connections between the probabilistically defined
quantity __\eqref{RJW-defU}__, gauges, and solutions of the reduced
Schrödinger equation __\eqref{RJW-Sch}__.
Besides continuing his own work in this area,
in the 1980s Chung had two students,
Elton P. Hsu
[5],
[e10],
[e11]
and Vassilus Papanicolaou
[e12],
who worked on probabilistic representations for other boundary-value problems
associated with the reduced Schrödinger equation.
A conjecture of Chung on equivalent conditions for finiteness of the
gauge in terms of
finiteness of __\( u(D,q, f; \,\cdot\,) \)__, when
__\( f \)__ is a non-negative
function that is positive only on a suitable subset of the boundary,
stimulated
my work
[e9]
(as a student of Chung) and then Neil Falkner’s
[e8]
(as a visitor at Stanford).
Falkner’s work used conditioned Brownian motion,
which became an object of intense interest, in its own right
and for its connections with gauge theorems.
For more on this fascinating subject, see the
accompanying article by Michael Cranston.
Other generalizations have also occurred, especially ones involving more
general Markov processes than Brownian motion.
The works related to this are too numerous to mention here.

Finally, on a personal note, I would like to thank Kai Lai Chung for the pleasure of our collaborations and for the many lively discussions I have enjoyed with him over the years.