Celebratio Mathematica

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.