#### by Michael Cranston

Through his works and words, Kai Lai Chung has been the spur for substantial developments in the understanding of
conditional Brownian motion and its application to the theory of
Schrödinger operators. Many in the field received mail or phone calls from Chung with interesting and provocative questions on the subject. At the Spring 1982 meeting of the Seminar on Stochastic Processes, he posed an interesting question on the lifetime of conditional Brownian motion. The resolution of this question (described below) has led to wide-ranging developments. His foundational work with Rao on the gauge theorem, to mention just one of his many works in this area, has served as the motivation for many developments in the understanding of
Schrödinger operators and their semigroups. And, of course,
*From Brownian Motion to Schrödinger’s Equation*
[1]
with Zhongxin Zhao has served as guide to developments in the field. In this short, semi-accurate, historical note, I’d like to outline a few developments that trace their origins to the encouragement of Chung. I’d like to apologize in advance for the many works which are not mentioned here, due in large part to an interest in brevity.

First, an introduction is in order. Let __\( B \)__ denote Brownian motion on __\( \mathbb{R}^d \)__ defined on the probability space __\( (\Omega, \mathcal F_t, \{P_x\}_{x\in \mathbb{R}^d}) \)__. For __\( D\subset \mathbb{R}^d \)__, let __\( p(t,x,y) \)__ be its transition density when killed at time
__\[ \tau^{}_D=\inf\{t > 0:B_t\notin D\} \]__
(the heat kernel on __\( D \)__.) Given a positive super-harmonic function __\( h \)__ on __\( D \)__, define
__\[ p^h(t,x,y)=\frac{p(t,x,y)\,h(y)}{h(x)} .\]__
This is the transition density for a new diffusion, called the __\( h \)__-process or conditional Brownian motion. We denote by __\( P^h_x \)__ the measure on path-space corresponding to Brownian motion started at __\( x \)__ and with transition density __\( p^h(t,x,y) \)__. In the case when __\( h \)__ is the Martin kernel with pole at the Martin boundary point __\( \xi \)__, then the conditional Brownian motion exits the domain at the boundary point __\( \xi \)__, in the sense that __\( B_t \)__ converges __\( P_x^h \)__-a.s. to __\( \xi \)__ in the Martin topology as __\( t \)__ approaches the path lifetime, __\( \tau^{}_D \)__. If __\( h(\,\cdot\,)=G_D(\,\cdot\,,y) \)__, where __\( G_D \)__ is the Green function for __\( D \)__ and __\( y\in D \)__, then the __\( h \)__-process will converge to __\( y \)__ as __\( t \)__ approaches the path lifetime. When __\( h \)__ is the Martin kernel with pole at __\( \xi \)__, we denote the resulting measure by __\( P^\xi_x \)__ and, when __\( h(\,\cdot\,)=G(\,\cdot\,,y) \)__, by __\( P_x^y \)__. These were developments due to
Doob
[e1]
in his study of probabilistic versions of the Fatou boundary-limit results for harmonic functions. Now it’s been known for some time that, if __\( D \)__ is bounded, then, for unconditioned Brownian motion,
__\begin{equation}\label{ceqn1}
E_x[\tau^{}_D] < c_d \operatorname{vol}(D)^{2/d},
\end{equation}__
and, if __\( \lambda_1 \)__ is the first Dirichlet eigenvalue for __\( \frac12\Delta \)__ on __\( D \)__, then
__\begin{equation}\label{ceqn2}
\lim_{t\rightarrow \infty}\frac{1}{t}\log P_x(\tau^{}_D > t)=-\lambda_1.
\end{equation}__
Using the Martin boundary, denoted here by __\( \partial_M D \)__, the expected lifetime can be expressed as
__\[ E_x[\tau^{}_D]=\int_{\partial_MD}E_x^\xi[\tau^{}_D]\,\omega_x(d\xi) ,\]__
where __\( \omega_x \)__ is the exit distribution of Brownian motion on __\( \partial_MD \)__, also known as the harmonic measure. So, by Fubini, __\( E_x^\xi[\tau^{}_D] \)__ is finite __\( \omega_x \)__-almost surely. Chung’s question is this:
__\[ \text{When is } E_x^\xi[\tau^{}_D] \text{ bounded uniformly in } x \text{ and } \xi? \]__
Or, more generally, when is __\( E_x^\xi[\tau^{}_D] \)__ finite?
This innocuous-sounding question turned out to have quite broad implications. It led to the introduction of some very interesting ideas from analysis into probability theory, such as the boundary Harnack principle, Whitney chains, Littlewood–Paley __\( g \)__-function and intrinsic ultracontractivity.

The first result on this question,
due to McConnell and the author
[e5],
was that there is a positive constant __\( c \)__ so that, if __\( D\subset \mathbb{R}^2 \)__ and __\( h \)__ is a positive harmonic function on __\( D \)__, then
__\begin{equation} \label{ceqn3}
E_x^h[\tau^{}_D]\le c\operatorname{vol}(D).
\end{equation}__
This is the analog then of __\eqref{ceqn1}__ in __\( d=2 \)__.
An example was given of a bounded __\( D\subset \mathbb{R}^3 \)__ with a __\( \xi\in \partial_MD \)__ for which __\( E_x^\xi[\tau^{}_D]=\infty \)__. Thus, the analog of __\eqref{ceqn1}__ cannot hold for __\( d > 2 \)__ without further assumptions. First, a word or two on the proof of __\eqref{ceqn3}__. This relies on decomposing the domain __\( D \)__ into subregions by means of the __\( 2^m \)__-level sets of the function __\( h \)__. That is,
__\[ D=\bigcup_{m=-\infty}^{\infty} D_m
\quad\text{ where }\quad D_m=\{x\in D:2^{m-1} < h(x) < 2^{m+1}\}
.\]__
The conditional Brownian motion viewed at the successive hitting times to
__\[ C_m=\{x\in D:h(x)=2^m\} \]__
forms a birth and death Markov chain on __\( \{2^m:m\in \mathbb{Z}\} \)__, with probability __\( 2/3 \)__ of going up and __\( 1/3 \)__ of going down. This implies that the number of visits to the __\( C_m \)__ are geometrically distributed random variables. These have finite expectation with a value independent of __\( m \)__. The other key observation is that the expected amount of time the conditional Brownian motion spends in __\( D_m \)__ starting from __\( C_m \)__ is equivalent (since __\( 1/2 \le h(y)/h(x) \le 2 \)__ for __\( x\in C_m \)__, __\( y \in D_m \)__) to the amount of time standard Brownian motion spends in __\( D_m \)__ starting from __\( C_m \)__. Combining this observation with __\eqref{ceqn1}__ gives that the expected time spent in __\( D_m \)__ starting on __\( C_m \)__ by the __\( h \)__-process is bounded by __\( C_d
\operatorname{vol}(D_m)^{2/d} \)__. Using the strong Markov property and summing leads to an upper bound of
__\[E_x^h[\tau^{}_D]\le C_d \sum_{m=-\infty}^{\infty}\operatorname{vol}(D_m)^{2/d}. \]__
In case __\( d=2 \)__, the sum is bounded by __\( 2\operatorname{vol}(D) \)__, leading to the result that there is a constant __\( c \)__ such that __\eqref{ceqn3}__ holds for
__\( D\subset \mathbb{R}^2 \)__, __\( x\in D \)__, and __\( h \)__ any positive superharmonic on __\( D \)__.
Since __\( 2/d < 1 \)__ for __\( d\ge 3 \)__, the finiteness of __\( \sum_{m=-\infty}^{\infty}\operatorname{vol}(D_m)^{2/d} \)__ does not generally hold, and leads to interesting questions about the influence of the regularity of the boundary and its effect on the size of the sets __\( D_m \)__. (The relation between boundary regularity and the growth of harmonic functions is a key issue in the subject.) This question was addressed by
Bañuelos
[e9],
Falkner
[e10],
Bass and Burdzy
[e23],
DeBlassie
[e13],
Kenig and Pipher
[e17],
and myself
[e7],
among others in the higher-dimensional case. The results of
Bañuelos
[e9]
incorporated many of the types of domains encountered in analysis, namely Lipschitz, NTA (non-tangentially accessible), John- and BMO-extension (uniform) domains.
In order to describe the results in
[e9],
we consider a Whitney decomposition of __\( D \)__. This is a collection of closed squares __\( Q_j \)__ with sides parallel to the coordinate axes and __\( D=\bigcup_j Q_j \)__ with the properties
__\begin{align*}
& Q_j^o\cap Q_k^o=\emptyset
\quad\text{if }j\neq k,\\
& \frac{1}{4}\le\frac{l(Q_j)}{l(Q_k)}\le 4
\quad\text{if } Q_j\cap Q_k\neq\emptyset,\\
& 1\le\frac{d(Q_j,\partial D)}{l(Q_j)}\le 4\sqrt d
\quad\text{for all }j.
\end{align*}__
A Whitney chain connecting __\( Q_j \)__ and __\( Q_k \)__ is a sequence of Whitney squares __\( \{Q_{m_i}\}_{i=0}^n \)__ with __\( Q_{m_0}=Q_j \)__, __\( Q_{m_n}=Q_k \)__ and
__\( Q_{m_i}\cap Q_{m_{i+1}}\neq \emptyset \)__.
An important fact about Whitney squares is that there is a positive constant __\( c \)__ so that, for any positive harmonic function __\( h \)__ in __\( D \)__ and adjacent Whitney squares
__\( Q_j\cap Q_k\neq \emptyset \)__,
we have __\( h(x) < c\,h(y) \)__ for __\( x\in Q_j \)__, __\( y\in Q_k \)__. Whitney chains are very well suited to the study of conditional Brownian motion. The reason is that, due to Harnack’s inequality, any positive harmonic function will be “flat” on the Whitney square __\( Q_j \)__. This means that the transition densities __\( p^h(t,x,y) \)__ and __\( p(t,x,y) \)__ will be equivalent on __\( Q_j \)__, which means the behavior of ordinary and conditional Brownian motion will be comparable on __\( Q_j \)__. Now, define the quasi-hyperbolic distance from __\( x\in Q_j \)__ to __\( x_0 \)__ by first setting __\( d(x)=\operatorname{dist}(x,\partial D) \)__ and then putting,
__\[\rho^{}_D(x_1,x_2)=\inf_\gamma \int_\gamma \frac{ds}{d(\gamma(s))},\]__
with the __\( \inf \)__ being taken over all rectifiable curves in __\( D \)__ from __\( x_1 \)__ to __\( x_2 \)__. Taking points __\( x_1\in Q_j \)__, __\( x_2\in Q_k \)__ we have
__\[\rho^{}_D(x_1,x_2)\,\approx\,
\text{length of shortest Whitney chain from }Q_j\text{ to }Q_k
.\]__
Note that repeated applications of Harnack’s inequality in successive squares in a Whitney chain implies that __\( h(x_1)\le c^{\rho^{}_D(x_1,x_2)}h(x_2) \)__. If we fix an __\( x_0\in D \)__ and write __\( \rho^{}_D(x)=\rho^{}_D(x,x_0) \)__, this implies that, for some constant __\( C \)__,
__\[ D_m\subset \{x\in D:\rho^{}_D(x) > C|m|\}. \]__
Now, a result of
Smith and Stegenga
[e19]
implies that, for a class __\( H(0) \)__ of domains, called Hölder of order 0 (which includes Lipschitz, NTA, John- and BMO-extension domains), one has __\( \rho^{}_D(\,\cdot\,,x_0)\in L^p(D) \)__ for any __\( 0 < p < \infty \)__. Using this, Bañuelos obtains for __\( D \)__ a bounded Hölder of order 0 domain so that
__\[ \sum_{m=-\infty}^{\infty}\operatorname{vol}(D_m)^{2/d} < \infty. \]__
This implies that __\( H(0) \)__ domains are regular enough so that an analog of __\eqref{ceqn3}__ holds for them in all dimensions.
There are also beautiful connections in simply connected planar domains between the behavior of conditional Brownian motion and the hyperbolic geometry of the region. This was developed in
Bañuelos and Carrol
[e25],
and
Davis
[e1].
We start our exposition of this connection with an observation of
Bañuelos
[e24].
If __\( D \)__ is a simply connected planar domain, and __\( \varphi: B(0,1)\rightarrow D \)__ maps the unit disc __\( B(0,1) \)__ of the complex plane conformally onto __\( D \)__ with __\( \varphi(0)=x \)__, then
__\[ g^2_*(\varphi)(\theta)=\frac{1}{\pi}\int_{B(0,1)}\log\Bigl(\frac{1}{|z|}\Bigr)\,\frac{1-|z|^2}{|z-e^{i\theta}|^2}\,|\varphi^{\prime}(z)|^2 \,dz \]__
is the Littlewood–Paley square function. Recalling that the Green function of __\( B(0,1) \)__ with pole at the origin is __\( \log(1/|z|) \)__, and that the Green function is preserved by conformal mappings, it is easy to deduce that, for __\( h \)__ a positive harmonic function on __\( D \)__ with the representation
__\[ h(z)=\int_0^{2\pi}\frac{1-|z|^2}{|z-e^{i\theta}|}\,d\mu(\theta) \]__
with __\( \mu \)__ a positive Borel measure on __\( \partial B(0,1) \)__, that
__\begin{align}\label{ceqn4}
E^h_x[\tau^{}_D]
&=\frac{1}{h(x)}\int_D G_D(x,y)\,h(y)\,dy
\\
&= \frac{1}{h(x)}\int_{B(0,1)}\log\Bigl(\frac{1}{|z|}\Bigr)\,h(\varphi(z))\,|\varphi^{\prime}(z)|^2\,dx\,dy
\nonumber
\\
&= \frac{1}{h(x)}\int_0^{2\pi}g_*^2(\varphi)(\theta)\,d\mu(\theta).
\nonumber
\end{align}__
Since __\( \mu([0,2\pi])=h(x) \)__ and
__\[ g^2_*(\varphi)(\theta)\le C\int_{B(0,1)}|\varphi^{\prime}(z)|^2\,dz \le C\operatorname{vol}(D), \]__
it follows that
__\[ E^h_x[\tau^{}_D]\le C\operatorname{vol}(D), \]__
thus giving another derivation of the lifetime estimate in the special case of simply connected planar domains.
But this gives additional information, as developed in
Bañuelos and Carrol
[e25].
There, the authors observed that, if __\( K(z,\xi) \)__ is the Poisson kernel for __\( B(0,1) \)__ with pole at __\( \xi\in\partial B(0,1) \)__, then there are positive constants __\( c \)__ and __\( C \)__ such that
__\[ c\sup_\varphi g^2_*(\varphi)(0)
\,\le\,
\sup_\varphi \int_{B(0,1)}\!\!\! K(z,1)K(z,-1)\,|\varphi^{\prime}(z)|^2\,dx\,dy
\,\le\,
C\sup_\varphi g^2_*(\varphi)(0), \]__
where the __\( \sup \)__ is taken over all conformal mappings __\( \varphi: B(0,1)\rightarrow D \)__ with __\( \varphi(0)=x \)__.
But another equivalence holds for __\( K(z,1)K(z,{-}1) \)__. Namely, if __\( d(z,\Gamma) \)__ denotes the hyperbolic distance in __\( B(0,1) \)__ from __\( z \)__ to the geodesic __\( \tilde{\Gamma}=[-1,1] \)__, then
__\[ \tfrac{1}{4}K(z,1)K(z,-1)\le e^{-2d(z,\tilde{\Gamma})}\le K(z,1)K(z,-1). \]__
Using the conformal invariance of the hyperbolic metric, writing __\( d^{}_D \)__ for the hyperbolic metric in __\( D \)__, and putting these two equivalences together yields the existence of two positive constants __\( c \)__ and __\( C \)__ such that
__\[ c\sup_{\Gamma}\int_De^{-2d^{}_D(z,\Gamma)}
\le \sup_{x,h}E_x^h[\tau^{}_D]
\le C\sup_{\Gamma}\int_De^{-2d^{}_D(z,\Gamma)}. \]__
This has a beautiful corollary involving the Whitney decomposition mentioned above. Let __\( Q \)__ be a Whitney cube with center __\( z^{}_Q \)__,
and let __\( T_Q \)__ be the total amount of time spent in __\( Q \)__ before __\( \tau^{}_D \)__. Then, for Martin boundary points __\( \xi_1,\xi_2 \)__ and for __\( \Gamma \)__ the hyperbolic geodesic connecting them, there are positive constants __\( c \)__ and __\( C \)__ such that
__\[ \tfrac{1}{4}e^{-Cd^{}_D(z^{}_Q,\Gamma)}\le E_{\xi_1}^{\xi_2}[T_Q]\le e^{-cd^{}_D(z^{}_Q,\Gamma)}. \]__
This is a quantitative statement about how closely the conditional Brownian motion from __\( \xi_1\,\text{to}\,\xi_2 \)__ follows the hyperbolic geodesic from __\( \xi_1\,\text{to}\,\xi_2 \)__.
Davis
[e14]
pursued this connection further in estimating the variance of __\( \tau^{}_D \)__ under the measure __\( E_{\xi_1}^{\xi_2} \)__. If __\( Q \)__ and __\( R \)__ are Whitney squares, then setting
__\[ P_Q=P_{\xi_1}^{\xi_2}(\tau^{}_{D\cap Q^c} < \tau^{}_D)
\quad\text{and}\quad
P_R=P_{\xi_1}^{\xi_2}(\tau^{}_{D\cap R^c} < \tau^{}_D) \]__
and letting __\( \delta(D) \)__ be the area of the largest disc which can be inscribed in __\( D \)__ yields
__\begin{align*}
& |\operatorname{Cov}_{\xi_1}^{\xi_2}(T_Q,T_R)|
\le Ce^{-c\delta^{}_D(z^{}_Q, z^{}_R)}
\operatorname{vol}(Q)
\operatorname{vol}(R)\,(P_Q+P_R)
\quad\text{and}
\\
& \operatorname{Var}_{\xi_1}^{\xi_2}(\tau^{}_D)
\le \delta(D)\,E_{\xi_1}^{\xi_2}[\tau^{}_D].
\end{align*}__
The first of these shows exactly how the decay of the dependence between the occupation times __\( T_Q \)__ and __\( T_R \)__ depends on the hyperbolic distance between __\( Q \)__ and __\( R \)__. The second confirms the intuition that the conditional Brownian motion speeds up when traversing narrow channels.
(If __\( D \)__ is a rectangle of length __\( n \)__ and width __\( 1/n \)__, then, for __\( {\xi_1} \)__ and __\( {\xi_2} \)__ on opposite ends of the long side of the rectangle,
__\[ E_{\xi_1}^{\xi_2}[\tau^{}_D]\le c
\quad\text{and}\quad
\operatorname{Var}_{\xi_1}^{\xi_2}[\tau^{}_D]\le c/n .\]__
Thus, the conditional motion must go a distance __\( n \)__ in a time with bounded expectation, independent of __\( n \)__, but with variance bounded by __\( 1/n \)__. This means the mass of the measure __\( P_{\xi_1}^{\xi_2} \)__ is concentrating on paths which make the length-__\( n \)__ trip in a time which is some constant that doesn’t depend on __\( n \)__.)

Refinements and further progress in these directions can be found in the works of Griffin, McConnell and Verchota [e27], Griffin, Verchota and Vogel [e26], Zhang [e30], Davis and Zhang [e28], and Xu [e20], to name but a few.

Now, let’s turn our attention to the problem of deciding to what extent the analog of __\eqref{ceqn2}__ holds for conditional Brownian motion. From the case of a ball __\( D=\{x:|x| < r\} \)__ in Euclidean space where
__\[ P_0(\tau^{}_D > t)=P_0^\xi(\tau^{}_D > t) \]__
for every boundary point __\( \xi \)__, one might suspect that, with some smoothness in __\( d > 2 \)__ and maybe even with __\( \operatorname{vol}(D) < \infty \)__ in __\( d=2 \)__, if __\( H^+(D) \)__ is the class of positive harmonic functions on __\( D \)__, then
__\begin{equation} \label{ceqn5}
\lim_{t\rightarrow \infty}\frac{1}{t}\log P^h_x(\tau^{}_D > t)=-\lambda_1
\quad\text{for }x\in D\text{ and }h\in H^+(D).
\end{equation}__
This was addressed in
De Blassie
[e13]
where it was proved that
__\eqref{ceqn5}__ holds provided __\( D \)__ is a Lipschitz domain with sufficiently small Lipschitz constant. Later,
Kenig and Pipher
[e17]
extended this result to Lipschitz domains and NTA domains. Perhaps the nicest approach is due to
Bañuelos
[e21]
and Bañuelos and Davis
[e15],
which illuminates the relation between the tail behavior of
the lifetime of conditional Brownian motion and intrinsic
ultracontractivity. The notion of intrinsic ultracontractivity
is defined
in
Davies and Simon
[e8]
as the property that the semigroup of the ground-state transformation of an operator maps __\( L^2 \)__ to __\( L^\infty \)__. To make this definition precise in the current setting, if __\( \varphi_1 \)__ is the first Dirichlet eigenfunction for __\( \frac{1}{2}\Delta \)__ on __\( D \)__, define a semigroup on __\( L^2(\varphi_1^2\,dx) \)__ by
__\[ P^{\varphi_1}_t f(x)
=\int_D\frac{e^{\lambda_1 t}p(t,x,y)}{\varphi_1(x)\varphi_1(y)}
f(y)\,\varphi_1^2(y)\,dy
\quad\text{for }f \in L^2(\varphi_1^2\,dx). \]__
Then, the domain __\( D \)__ is defined to be intrinsically ultracontractive (IU) if there exist constants __\( C_t \)__ such that
__\[ |P^{\varphi_1}_t f(x)|\le C_t\|f\|_{L^2(\varphi_1^2dx)}
\quad\text{for }t > 0 .\]__
An important consequence of IU is that for any __\( \varepsilon > 0 \)__ there is a __\( t(\varepsilon) \)__ such that
__\begin{equation}\label{ceqn6}
(1-\varepsilon)\,e^{-\lambda_1t}\varphi_1(x)\varphi_1(y)
\le p(t,x,y)
\le(1+\varepsilon)\,e^{-\lambda_1t}\varphi_1(x)\varphi_1(y).
\end{equation}__
Since, for any __\( h\in H^+(D) \)__,
__\[ P_x^h(\tau^{}_D\ge t)=\frac{1}{h(x)}\int_Dp(t,x,y)h(y)\,dy \le 1,\]__
it follows easily from __\eqref{ceqn5}__
that
__\[ \lim_{t \rightarrow \infty}\frac{1}{t}\log P_x^h(\tau^{}_D\ge t)=-\lambda_1,\]__
giving the Bañuelos analog of __\eqref{ceqn2}__ for conditional Brownian motion on IU domains.
In the case of planar domains of finite area,
Bañuelos and Davis
[e15]
proved the following analog of IU for each __\( x\in D \)__:
__\[ \lim_{t\rightarrow \infty}\frac{e^{\lambda_1t}p(t,x,y)}{\varphi_1(x)\varphi_1(y)}=1 \text{ uniformly in }y\in D. \]__
This implies that the analog of __\eqref{ceqn2}__ for conditional Brownian motion holds for planar domains of finite area.

Another application of conditional Brownian motion, which has been an area of research to which Professor Chung has made many contributions, is to the study of the
Schrödinger equation by means of the Feynman–Kac formula. A seminal paper on the subject was that of
Aizenman and Simon
[e2],
who used path-integral techniques (the Feynman–Kac formula) to prove Harnack’s inequality for
Schrödinger operators. Consider, with __\( d > 2 \)__ for ease of presentation, a potential __\( V \)__ satisfying
__\[ \lim_{r \rightarrow 0}\sup_{x \in \mathbb{R}^d}\int_{|x-y| < r}\frac{|V(y)|\ }{|x-y|^{d-2}}\,dy =0. \]__
The class of such potentials is called the Kato class, and is denoted by __\( K_d \)__. They are particularly well suited to the Newtonian potential, and thus as well to the occupation properties of Brownian motion. Now, for __\( f\in C(\partial D) \)__, consider the Dirichlet problem
__\begin{align} \label{ceqn7}
\tfrac{1}{2}\Delta u(x)+V(x)u(x) &=0
\quad\text{for }x\in D,\\
u(x) &=f(x)
\quad\text{for }x\in \partial D.
\nonumber
\end{align}__
The Gauge Theorem of Chung and Rao
(see the article of Ruth Williams in this volume)
says that either
__\( E_x\bigl[e^{\int_0^{\tau^{}_D}V(B_s)\,ds}\bigr]\equiv \infty \)__ on __\( D \)__,
or this quantity (called the gauge) is bounded on __\( D \)__. Let’s assume that the second alternative of this dichotomy holds. Then, by Feynman–Kac, the solution of
__\eqref{ceqn7}__
is given by
__\begin{eqnarray*}
u(x)= E_x\bigl[e^{\int_0^{\tau^{}_D}V(B_s)\,ds}f(B_{\tau^{}_D})\bigr].
\end{eqnarray*}__

Let’s suppose now that __\( D \)__ is a Lipschitz domain so that the Euclidean and Martin boundary of __\( D \)__ are the same. Decompose the Feynman–Kac formula using conditional Brownian motion,
__\begin{equation}\label{ceqn8}
u(x)= \int_{\partial D}
E_x^y\bigl[e^{\int_0^{\tau^{}_D}V(B_s)\,ds}\bigr]
f(y)\,P_x(B_{\tau^{}_D}\in dy).
\end{equation}__
The analog of Chung’s question regarding the finiteness of the expected lifetime of conditional Brownian motion, as well as his question regarding the finiteness of the gauge, is:
__\[ \text{When is }\,E_x^y\bigl[e^{\int_0^{\tau^{}_D}V(B_s)\,ds}\bigr] < \infty\,\text{?} \]__
The quantity
__\[ u(x,y)=E_x^y\bigl[e^{\int_0^{\tau^{}_D}V(B_s)\,ds}\bigr] \]__
is known as the conditional gauge. Under the conditions set down above, namely that __\( D \)__ be a Lipschitz domain and __\( V\in K_d \)__, a dichotomy (similar to the Gauge Theorem) holds:
either
__\[ E_x^y\bigl[e^{\int_0^{\tau^{}_D}V(B_s)\,ds}\bigr]\equiv \infty\]__
or there are positive constants __\( c \)__ and __\( C \)__ such that
__\begin{equation} \label{ceqn9}
c\le E_x^y\bigl[e^{\int_0^{\tau^{}_D}V(B_s)\,ds}\bigr]\le C
\quad\text{for all }x,y\in D \cup \partial D.
\end{equation}__
This is called the Conditional Gauge Theorem (CGT). It can be viewed as a statement on the mixing properties of conditional Brownian motion. The potential __\( V \)__ may possess singularities. The CGT says that these singularities can’t be so bad that __\( P_x^y \)__-paths would miss them, in the sense that
__\[ E_x^y\bigl[e^{\int_0^{\tau^{}_D}V(B_s)\,ds}\bigr] < \infty \]__
for one pair of points __\( x,y \)__, but
__\[ E_z^w\bigl[e^{\int_0^{\tau^{}_D}V(B_s)\,ds}\bigr]=\infty \]__
for another pair __\( z,w \)__.
That is, under both measures __\( P_x^y \)__ and __\( P_z^w \)__, the occupation distributions of paths are similar enough that they will simultaneously give a finite answer or an infinite answer when asked about the value of the conditional gauge. This requires some smoothness of __\( \partial D \)__ with its resulting effect on the behavior of the Green function. Early results on the subject were those of
Falkner
[e4]
and Zhao
[e3],
[e6].
In the fundamental works of Zhao, the CGT was proved for Kato-class potentials on the ball, and then domains with __\( C^2 \)__ boundary. For Lipschitz domains and Kato potentials, the result was proven in
Cranston, Fabes and Zhao
[e12].
The extension to Lipschitz domains of the CGT used the so-called 3__\( G \)__-theorem. This result says that, if __\( G \)__ is the Green function for __\( \tfrac{1}{2}\Delta \)__ on __\( D \)__, then there is a positive constant __\( C \)__ such that
__\begin{equation} \label{ceqn10}
\frac{G(x,z)G(z,y)}{G(x,y)}\le
C\Bigl(\frac{1}{|x-z|^{d-2}}+\frac{1}{|y-z|^{d-2}}\Bigr).
\end{equation}__
The left-hand side in __\eqref{ceqn10}__ is the Green function for
conditional Brownian motion started at __\( x \)__ and conditioned to exit __\( D \)__ at __\( y \)__. This is the occupation density for conditional Brownian motion in __\( D \)__, in the sense that the total expected amount of time spent by __\( B \)__ in __\( A\subset D \)__ with respect to the measure __\( P_x^y \)__ is
__\[ \int_A \frac{G(x,z)G(z,y)}{G(x,y)}\,dz .\]__
The right-hand side of __\eqref{ceqn10}__ is the sum of the Newtonian potentials with poles at __\( x \)__ and __\( y \)__,
respectively. These are the occupation densities for unconditioned Brownian motion in __\( \mathbb{R}^d \)__ started at __\( x \)__ and __\( y \)__. The 3__\( G \)__-inequality says that, if __\( V \)__ is in __\( K_d \)__ and thus well adapted to the occupation measure of (unconditioned) Brownian motion, then it is also well adapted to the occupation measure of conditional Brownian motion. In the case when the conditional gauge is finite, the CGT permits comparisons between potential theoretic quantities for the two operators __\( -\frac{1}{2}\Delta \)__ and __\( -\frac{1}{2}\Delta+ V \)__. This lies close to the original motivation of
Aizenman and Simon
[e2].
For example, suppose that for some __\( f\in C(\partial D) \)__,
__\begin{align*}
\tfrac{1}{2}\Delta v(x) &=0 \quad\text{for } x \in D,\\
v(x) &=f(x) \quad\text{for } x\in \partial D.
\end{align*}__
and
__\begin{align*}
-\tfrac{1}{2}\Delta u(x)+V(x)u(x) &=0 \quad\text{for } x \in D,\\
u(x) &=f(x) \quad\text{for } x\in \partial D.
\end{align*}__
Then, __\( v(x)= E_x[f(B_{\tau^{}_D})] \)__ and, since
__\[ c\le E_x^y[e^{\int_0^{\tau^{}_D}V(B_s)\,ds}]\le C ,\]__
it follows from __\eqref{ceqn4}__ that __\( cv(x)\le u(x)\le Cv(x),\,x \in D \)__. With this equivalence, Harnack’s inequality, and even the boundary Harnack inequality, can be deduced for positive solutions in __\( D \)__ of
__\[ -\tfrac{1}{2}\Delta u(x)+V(x)u(x)=0 .\]__
Many other similar conclusions follow in an equally easy manner. Using the simple formula
__\begin{equation} \label{ceqn11}
u(x,y)=\frac{G_V(x,y)}{G(x,y)},
\end{equation}__
it follows that
__\[cG(x,y)\le G_V(x,y) \le CG(x,y)
\quad\text{for }x,y\in D,\]__
where __\( G_V \)__ is the Green function for __\( -\frac{1}{2}\Delta+ V \)__.
Since the Martin kernels __\( K(x,\xi) \)__ and __\( K_V(x,\xi) \)__ are the limit of ratios of the Green functions __\( G(x,\xi) \)__ and __\( G_V(x,\xi) \)__, it follows as well that
__\[ cK(x,y)\le K_V(x,y) \le CK(x,y)
\quad\text{for } x,y\in D,\]__
where __\( K \)__ and __\( K_V \)__ are the Martin kernels for __\( \frac{1}{2}\Delta \)__ and __\( -\frac{1}{2}\Delta+ V \)__, respectively. Two-dimensional versions of these results appeared in
Bass and Burdzy
[e29],
Cranston
[e16],
McConnell
[e18],
and Zhao
[e11].
Results similar in flavor and which also incorporate the notion of IU above are due to
Bañuelos
[e24],
who proved that, when the conditional gauge is finite and __\( D \)__ is a Lipschitz or NTA domain, there exist positive constants __\( c_t \)__ and __\( C_t \)__ such that
__\[c_tp(t,x,y)\le p_V(t,x,y)\le C_tp(t,x,y)
\quad\text{for }t > 0\text{ and }x,y\in D, \]__
where __\( p_V \)__ is the heat kernel for __\( -\frac{1}{2}\Delta+V \)__.
An additional result of Bañuelos in this connection is that, if the conditional gauge is finite and __\( D \)__ is an __\( H(0) \)__ domain (as described earlier), then the operator __\( -\frac{1}{2}\Delta+ V \)__ is IU. It’s interesting to note that the proofs used log-Sobolev inequalities.
Further developments appear in a series of papers by
Chen and Song
[◊],
and Chen
[e32],
among others.
In
[e31],
the authors follow the developments of
Bañuelos
[e24],
and consider the conditional gauge problem for the fractional Laplacian, __\( (-\Delta)^\alpha \)__ for __\( 0 < \alpha < 2 \)__, and potentials in the suitably modified Kato class __\( K_{\alpha,d} \)__, where __\( V\in K_{\alpha,d} \)__ if
__\[\lim_{r \rightarrow 0}\sup_{\{x\in \mathbb{R}^d\}}
\int_{|x-y| < r}\frac{|V(y)|\ }{\ |x-y|^{d-\alpha}} \,dy=0. \]__
In this paper, Chen and Song
[e31]
deduced the CGT on __\( C^{1,1} \)__ domains for the operator __\( (-\Delta)^\alpha \)__ and __\( K_{\alpha,d} \)__ potentials. The proper process to use in the Feynman–Kac representation in this case is the symmetric stable process of order __\( \alpha \)__, __\( X \)__, rather than the Brownian motion used when considering __\( \Delta \)__. Their approach was to split the potential, writing __\( V=V_1+V_2 \)__ for __\( V_2\in \mathbf{L}^\infty \)__ and __\( V_1 \)__ with a small Kato norm, that is, with small
__\[ \sup_{x\in D}\int_D \frac{|V_1(y)|}{\ |x-y|^{d-\alpha}}dy .\]__
Then, by a simple lemma of Khasminski, they show that the Green functions __\( G^\alpha_{V_1} \)__ for __\( (-\Delta)^\alpha+V_1 \)__ and __\( G^\alpha \)__ for __\( (-\Delta)^\alpha \)__ on __\( D \)__ satisfy __\( G^\alpha_{V_1}\approx G^\alpha \)__, in the sense that there are positive constants __\( c \)__ and __\( C \)__ such that __\( cG^\alpha\le G^\alpha_{V_1}\le C G^\alpha \)__. This equivalence can then be used to prove that __\( (-\Delta)^\alpha+V \)__ is IU. However, __\( (-\Delta)^\alpha+V \)__ being IU implies that __\( G^\alpha \approx G^\alpha_{V} \)__. Now, using a formula analogue to __\eqref{ceqn7}__, the finiteness of the right-hand side follows from the inequality __\( G^\alpha_{V_1}\le C G^\alpha \)__. From
the 3__\( G \)__-Theorem for the Green function __\( G^\alpha \)__ on __\( D \)__, the CGT follows. This was extended in
[e33]
to __\( H(0) \)__ domains, again by an approach inspired by
[e24].

Relations between subcriticality and boundedness of the conditional gauge have been investigated by
Zhao
[e22].
Consider the class
__\[ B_c=\{q:\mathbb{R}^d\to\mathbb{R}:\operatorname{supp}q \text{ is compact}\}
\cap \mathbf{L}^\infty .\]__
The operator __\( -\frac{1}{2}\Delta +V \)__ is called subcritical if
__\[\text{for all }q\in B_c \text{ there exists } \varepsilon > 0
\text{ such that } -\tfrac{1}{2}\Delta +V+\varepsilon q\ge 0 . \]__
This amounts to a strict positivity of __\( -\frac{1}{2}\Delta +V \)__.

For a subclass of __\( K_d \)__ potentials which satisfy a condition at __\( \infty \)__,
Zhao
[e22]
proved that subcriticality is equivalent to
__\[ u(x,y)=E_x^y\bigl[e^{\int_0^\infty V(B_s)\,ds}\bigr]
\text{ is bounded on } \mathbb{R}^d\times \mathbb{R}^d. \]__
There were many other equivalences in that work which go a long way toward establishing the power of the approach in investigating the Schrödinger operator.

Generalizations of the Conditional Gauge Theorem to broader classes of Markov processes and potentials, including measures, have been carried out in Chen and Song [e31] and Chen [e32]. In the last work, Chen has proved gauge and conditional gauge theorems for a new class of Kato potentials, which even includes singular measures and general transient Borel right processes. And, most strikingly, following a suggestion of Chung, he proved that the CGT is actually the Gauge Theorem for the conditional process!

In this review, we’ve examined some of the many results which have connections with the works of Chung to be found in this volume. While we haven’t explicitly drawn the connections, we hope that these ties will become obvious to any reader of this volume. Finally, the author would like to express his gratitude to Professor Chung for introducing him to the fascinating problems in this area.