Celebratio Mathematica

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.

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 \( q \).

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:

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.

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 \( d=1 \),  \( D=(-\infty, \infty) \) 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].

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.

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).

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.