# Celebratio Mathematica

## Kai Lai Chung

### Stopped Feynman–Kac functionals and the Schrödinger equation

#### by Ruth Williams

In the late 1970s, Kai Lai Chung began in­vest­ig­at­ing con­nec­tions between prob­ab­il­ity and the (re­duced) Schrödinger equa­tion: $$\label{RW-reds} \tfrac{1}{2} \Delta u(x) + q(x)u(x) = 0 \quad \hbox{for } x\in \mathbb{R}^d,$$ 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: $$\label{RW-teds} \frac{\partial \psi (t,x)}{\partial t} = \frac {1}{2} \Delta \psi(t,x) + q(x) \psi(t, x) \quad\text{for } t > 0, x\in \mathbb{R}^d,$$ with ini­tial con­di­tion $\psi(0,x) = g(x)$ for $x\in\mathbb{R}^d$, and where $$\Delta \psi (t,x) =\sum_{i=1}^d \frac{\partial^2 \psi}{\partial x_i^2} (t,x),$$ then it can be shown (for ex­ample, by us­ing Itô’s for­mula), that $$\label{RW-expk} \psi(t, x) = E^x \bigl[ e_q(t) g(B_t)\bigr] \quad\text{for } t\geq 0, x\in \mathbb{R}^d.$$ 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 $$\label{RW-fkX} E^x\Bigl[\exp\Bigl(\int_0^{\tau} q(X_s)\,ds\Bigr) f(X_{\tau})\Bigr] \quad\text{for } x\in \overline D,$$ 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 $$\label{RW-diffreds} {\mathcal L} u(x) + q(x) u(x) = 0 \quad \hbox{for } x\in D,$$ 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, $$\label{RW-FKS} e_q(\tau) = \exp\Bigl( \int_0^{\tau} q(X_s)\,ds\Bigr),$$ 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 $b\in \mathbb{R}$,

1. if $v(x, b) < \infty$ for some $x < b$, then $v(x, b) < \infty$ for all $x < b$,

2. 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.

1. $\beta =+\infty$.

2. $\alpha =-\infty$.

3. 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: $$\label{RW-schrod} \tfrac{1}{2} u^{\prime\prime}(x) + q u(x) =0 \quad\text{for } x \in (-\infty , b),$$ with the bound­ary con­di­tion $u(b) = 1$.

The fol­low­ing con­di­tions are equi­val­ent.

1. There is a twice con­tinu­ously dif­fer­en­ti­able, strictly pos­it­ive func­tion $u$ sat­is­fy­ing \eqref{RW-schrod} with $b=+\infty$.

2. $\beta =+\infty$.

3. $\alpha =-\infty$.

4. For all $a, b\in \mathbb{R}$, $$\label{RW-uab} v(a,b)\,v(b,a) \leq 1.$$

5. 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 $$\label{RJW-defU} u(D, q, f; x) = E^x \Bigl[ \exp\Bigl( \int_0^{\tau^{}_D } q(B_s)\,ds\Bigr) f(B_{\tau^{}_D}) ; \tau^{}_D < \infty\Bigr] \quad\text{for } x\in \overline D.$$ 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 $$\label{RJW-Kato} \lim_{\alpha \downarrow 0} \Bigl[ \sup_{x\in \mathbb{R}^d} \int_{|y-x|\leq \alpha} \bigl|G(x-y) q(y) \bigr| \,dy \Bigr ] =0,$$ 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 $D$, $$\label{RJW-Sch} \tfrac{1}{2} \Delta u(x) + q(x) u(x) = 0 \quad \hbox{for } x\in D,$$ 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.

### Works

[1]K. L. Chung and S. R. S. Varadhan: “Kac func­tion­al and Schrödinger equa­tion,” Stu­dia Math. 68 : 3 (1980), pp. 249–​260. MR 599148 Zbl 0448.​60054 article

[2]K. L. Chung: “On stopped Feyn­man–Kac func­tion­als,” pp. 347–​356 in Sémin­aire de prob­ab­il­ités XIV (Par­is, 1978–1979). Edi­ted by J. Azéma and M. Yor. Lec­ture Notes in Math­em­at­ics 784. Spring­er (Ber­lin), 1980. MR 580141 Zbl 0444.​60061 incollection

[3]K. L. Chung and K. M. Rao: “Feyn­man–Kac func­tion­al and the Schrödinger equa­tion,” pp. 1–​29 in Sem­in­ar on stochast­ic pro­cesses, 1981 (North­west­ern Uni­versity, Evan­ston, IL, April 1981). Edi­ted by E. Çin­lar, K. L. Chung, and R. K. Getoor. Pro­gress in Prob­ab­il­ity and Stat­ist­ics 1. Birkhäuser (Bo­ston, MA), 1981. MR 647779 Zbl 0492.​60073 incollection

[4]K. L. Chung: Lec­tures from Markov pro­cesses to Browni­an mo­tion. Grundlehren der Math­em­at­ischen Wis­senschaften 249. Spring­er (New York), 1982. MR 648601 Zbl 0503.​60073 book

[5]K. L. Chung and P. Hsu: “Gauge the­or­em for the Neu­mann prob­lem,” pp. 63–​70 in Sem­in­ar on stochast­ic pro­cesses, 1984 (North­west­ern Uni­versity, Evan­ston, IL, 1984). Edi­ted by E. Çin­lar, K. L. Chung, and R. K. Getoor. Pro­gress in Prob­ab­il­ity and Stat­ist­ics 9. Birkhäuser (Bo­ston, MA), 1986. MR 896722 Zbl 0585.​60064 incollection

[6]K. L. Chung and Z. X. Zhao: From Browni­an mo­tion to Schrödinger’s equa­tion. Grundlehren der Math­em­at­ischen Wis­senschaften 312. Spring­er (Ber­lin), 1995. MR 1329992 Zbl 0819.​60068 book