Celebratio Mathematica

In the late 1970s, Kai Lai Chung began in­vest­ig­at­ing con­nec­tions between prob­ab­il­ity and the (re­duced) Schrödinger equa­tion: $$\begin{array}{}\text{(1)}& \frac{1}{2}\mathrm{\Delta }u(x)+q(x)u(x)=0\text{for }x\in {\mathbb{R}}^d,\end{array}$$ where $q$ is a real-val­ued Borel-meas­ur­able func­tion on ${\mathbb{R}}^d$, and $\mathrm{\Delta }$ 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.

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 $q\le 0$, the fol­low­ing mul­ti­plic­at­ive func­tion­al of one-di­men­sion­al Browni­an mo­tion $B$: $$e_q(t)=\exp ({\int }_0^{t}q(B_s)ds)\text{for }t\ge 0,$$ 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 $\psi :[0,\mathrm{\infty })\times {\mathbb{R}}^d\to \mathbb{R}$ 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 $i,j=1,\dots ,d$, defined on $(0,\mathrm{\infty })\times {\mathbb{R}}^d$, and such that the fol­low­ing time-de­pend­ent Schrödinger equa­tion holds: $$\begin{array}{}\text{(2)}& \frac{\partial \psi (t,x)}{\partial t}=\frac{1}{2}\mathrm{\Delta }\psi (t,x)+q(x)\psi (t,x)\text{for }t>0,x\in {\mathbb{R}}^d,\end{array}$$ with ini­tial con­di­tion $\psi (0,x)=g(x)$ for $x\in {\mathbb{R}}^d$, and where $$\begin{array}{}\text{(3)}& \mathrm{\Delta }\psi (t,x)=\sum _{i=1}^d\frac{{\partial }^2\psi }{\partial x_i^2}(t,x),\end{array}$$ then it can be shown (for ex­ample, by us­ing Itô’s for­mula), that $$\begin{array}{}\text{(4)}& \psi (t,x)=E^x[e_q(t)g(B_t)]\text{for }t\ge 0,x\in {\mathbb{R}}^d.\end{array}$$ 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 $\text{(2)}$ when $d=1$ and $q\le 0$. However, rather than con­sid­er­ing this equa­tion dir­ectly, he worked with a re­duced Schrödinger equa­tion sim­il­ar to $\text{(1)}$ (with $q-s$ in place of $q$, and $d=1$) ob­tained by form­ally tak­ing the Laplace trans­form (with para­met­er $s$) in equa­tion $\text{(2)}$ 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 $\text{(4)}$ 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 $$\begin{array}{}\text{(5)}& E^x[\exp ({\int }_0^{\tau }q(X_s)ds)f(X_{\tau })]\text{for }x\in \bar{D},\end{array}$$ 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 ${\mathbb{R}}^d$,  $\bar{D}$ is the clos­ure of $D$,  $f$ is a con­tinu­ous func­tion defined on the bound­ary of $D$, and $q\le 0$ is Hölder-con­tinu­ous and bounded on $D$. Here, $P^x$ and $E^x$ de­note re­spect­ively prob­ab­il­ity and ex­pect­a­tion op­er­at­ors for $X$ start­ing from $x\in \bar{D}$. The do­main $D$ is reg­u­lar if $$P^x(\tau =0)=1\text{for each }x\in \partial D.$$ Un­der suit­able as­sump­tions on $X$ and as­sum­ing that $D$ is reg­u­lar, Dynkin showed that ex­pres­sions of the form $\text{(5)}$ yield con­tinu­ous func­tions on $\bar{D}$ that sat­is­fy the equa­tion $$\begin{array}{}\text{(6)}& \mathcal{L}u(x)+q(x)u(x)=0\text{for }x\in D,\end{array}$$ with con­tinu­ous bound­ary val­ues giv­en by $f$, where $\mathcal{L}$ is the in­fin­ites­im­al gen­er­at­or of the dif­fu­sion pro­cess $X$. The as­sump­tion that $q\le 0$ im­plies that the ac­tion of the stopped Feyn­man–Kac func­tion­al, $$\begin{array}{}\text{(7)}& e_q(\tau )=\exp ({\int }_0^{\tau }q(X_s)ds),\end{array}$$ 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 $\text{(5)}$ 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 $q\ge 0$) the ex­pres­sion in $\text{(5)}$ is fi­nite for all $x\in \bar{D}$ if and only if there is a con­tinu­ous func­tion $u$ that is strictly pos­it­ive on $\bar{D}$ and sat­is­fies $\text{(6)}$.

It was not un­til the work of Chung [2] that prob­ab­il­ist­ic solu­tions of $\text{(1)}$ 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 $\text{(7)}$ 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 $q$.

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 $\text{(7)}$ with bounded Borel-meas­ur­able $q:\mathbb{R}\to \mathbb{R}$, and $$\tau ={\tau }_b\equiv inf\{t>0:X_t=b\}\text{for }b\in \mathbb{R}.$$ As­sume that $P^x({\tau }_b<\mathrm{\infty })=1$ for each $x\in \mathbb{R}$ and $b\in \mathbb{R}$, and define $$v(x,b)=E^x[\exp ({\int }_0^{{\tau }_b}q(X_s)ds)]\text{for }x\in \mathbb{R},b\in \mathbb{R}.$$ Two fun­da­ment­al prop­er­ties of $v$ are that $0<v(x,b)\le \mathrm{\infty }$ 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:

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

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.

The equi­val­ence of (a), (b) and (c) is an ana­logue of Khas­m­in­skii’s [e4] res­ults, but with $d=1$,  $D=(-\mathrm{\infty },\mathrm{\infty })$ 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 ($d\ge 1$), $P^x$ 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 $\bar{D}$ and bound­ary $\partial D$, $m$ is $d$-di­men­sion­al Le­besgue meas­ure, and $f:\partial D\to \mathbb{R}$ is a bounded Borel-meas­ur­able func­tion with $f\ge 0$. Let $${\tau }_D^{}=inf\{t>0:B_t\notin D\},$$ be the first exit time of $B$ from $D$. Define $$\begin{array}{}\text{(10)}& u(D,q,f;x)=E^x[\exp ({\int }_0^{{\tau }_D^{}}q(B_s)ds)f(B_{{\tau }_D^{}});{\tau }_D^{}<\mathrm{\infty }]\text{for }x\in \bar{D}.\end{array}$$ When $m(D)<\mathrm{\infty }$,  $P^x({\tau }_D^{}<\mathrm{\infty })=1$ for all $x\in \bar{D}$ (see [6], The­or­em 1.17) and the qual­i­fi­er ${\tau }_D^{}<\mathrm{\infty }$ in $\text{(10)}$ may be omit­ted. The fol­low­ing is The­or­em 1.2 in Chung and Rao [3].

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 $q:{\mathbb{R}}^d\to \bar{\mathbb{R}}$ is a Borel-meas­ur­able func­tion sat­is­fy­ing $$\begin{array}{}\text{(11)}& \underset{\alpha \downarrow 0}{lim}[\underset{x\in {\mathbb{R}}^d}{sup}{\int }_{|y-x|\le \alpha }|G(x-y)q(y)|dy]=0,\end{array}$$ where $\bar{\mathbb{R}}=[-\mathrm{\infty },\mathrm{\infty }]$ is the ex­ten­ded real line and, for $x\in {\mathbb{R}}^d$, $$G(x)=\{\begin{array}{ll}|x{|}^{2-d}& \text{if }d\ge 3,\\ \ln {\displaystyle \frac{1}{|x|}}& \text{if }d=2,\\ |x|& \text{if }d=1.\end{array}$$ The set of Borel-meas­ur­able func­tions $q:{\mathbb{R}}^d\to \bar{\mathbb{R}}$ sat­is­fy­ing $\text{(11)}$ 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 $\text{(12)}$, 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 $\text{(12)}$ 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 $D$, $$\begin{array}{}\text{(12)}& \frac{1}{2}\mathrm{\Delta }u(x)+q(x)u(x)=0\text{for }x\in D,\end{array}$$ 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 $d\ge 2$, 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 $$|h(x)-h(y)|\le M|x-y{|}^{\alpha }\text{for all }x,y\in K.$$ The fol­low­ing the­or­em is proved in Chung and Rao [3] for $d\ge 2$. 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 $d=1$,  $m(D)<\mathrm{\infty }$ im­plies that $D$ is a bounded in­ter­val.

This the­or­em has been gen­er­al­ized to situ­ations where $q$ is a Kato-class func­tion, and $\text{(12)}$ 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 $f\ge 0$, then $u(D,q,f;\cdot )$ is a non-neg­at­ive solu­tion of $\text{(12)}$ and, if $f>0$ on $\partial D$, then $u(D,q,f;\cdot )>0$ on $\bar{D}$. One nat­ur­ally ex­pects there to be some re­la­tion between the ex­ist­ence of such pos­it­ive solu­tions of $\text{(12)}$ and the sign of $$\lambda (D,q)=\underset{\phi }{sup}[{\int }_D\{-\frac{1}{2}|\mathrm{\nabla }\phi (x){|}^2+q(x)\phi (x{)}^2\}dx],$$ where the su­prem­um is over all $\phi :D\to \mathbb{R}$ such that $\phi$ is in­fin­itely con­tinu­ously dif­fer­en­ti­able in $D$, has com­pact sup­port in $D$, and sat­is­fies ${\int }_D\phi (x{)}^{2}dx=1$. The quant­ity $\lambda (D,q)$ is the su­prem­um of the spec­trum of the op­er­at­or $\frac{1}{2}\mathrm{\Delta }+q$ 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).

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 $\text{(12)}$, 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 $\text{(10)}$, gauges, and solu­tions of the re­duced Schrödinger equa­tion $\text{(12)}$. 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.