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 $(\mathrm{\Omega },{\mathcal{F}}_t,\{P_x{\}}_{x\in {\mathbb{R}}^d})$. 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 $P_x^h$ 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 $P_x^{\xi }$ 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, $$\begin{array}{}\text{(1)}& E_x[{\tau }_D^{}]<c_d\mathrm{vol}(D{)}^{2/d},\end{array}$$ and, if ${\lambda }_1$ is the first Di­rich­let ei­gen­value for $\frac{1}{2}\mathrm{\Delta }$ on $D$, then $$\begin{array}{}\text{(2)}& \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}\log P_x({\tau }_D^{}>t)=-{\lambda }_1.\end{array}$$ Us­ing the Mar­tin bound­ary, de­noted here by ${\partial }_{M}D$, the ex­pec­ted life­time can be ex­pressed as $$E_x[{\tau }_D^{}]={\int }_{{\partial }_{M}D}{E}_x^{\xi }[{\tau }_D^{}]{\omega }_x(d\xi ),$$ where ${\omega }_x$ is the exit dis­tri­bu­tion of Browni­an mo­tion on ${\partial }_{M}D$, 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 $$\begin{array}{}\text{(3)}& E_x^h[{\tau }_D^{}]\le c\mathrm{vol}(D).\end{array}$$ This is the ana­log then of $\text{(1)}$ in $d=2$. An ex­ample was giv­en of a bounded $D\subset {\mathbb{R}}^3$ with a $\xi \in {\partial }_{M}D$ for which $E_x^{\xi }[{\tau }_D^{}]=\mathrm{\infty }$. Thus, the ana­log of $\text{(1)}$ can­not hold for $d>2$ without fur­ther as­sump­tions. First, a word or two on the proof of $\text{(3)}$. This re­lies on de­com­pos­ing the do­main $D$ in­to sub­re­gions by means of the $2^m$-level sets of the func­tion $h$. That is, $$D=\bigcup _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{D}_m\text{ where }{D}_m=\{x\in D:2^{m-1}<h(x)<2^{m+1}\}.$$ 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 $\text{(1)}$ gives that the ex­pec­ted time spent in $D_m$ start­ing on $C_m$ by the $h$-pro­cess is bounded by $C_d\mathrm{vol}(D_m{)}^{2/d}$. Us­ing the strong Markov prop­erty and sum­ming leads to an up­per bound of $$E_x^h[{\tau }_D^{}]\le C_d\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{vol}(D_m{)}^{2/d}.$$ In case $d=2$, the sum is bounded by $2\mathrm{vol}(D)$, lead­ing to the res­ult that there is a con­stant $c$ such that $\text{(3)}$ 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 $\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{vol}(D_m{)}^{2/d}$ 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 $$\begin{array}{rl}& Q_j^o\cap Q_k^o=\mathrm{\varnothing }\text{if }j\ne k,\\ & \frac{1}{4}\le \frac{l(Q_j)}{l(Q_k)}\le 4\text{if }{Q}_j\cap Q_k\ne \mathrm{\varnothing },\\ & 1\le \frac{d(Q_j,\partial D)}{l(Q_j)}\le 4\sqrt{d}\text{for all }j.\end{array}$$ 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 $Q_{m_i}\cap Q_{m_{i+1}}\ne \mathrm{\varnothing }$. 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 $Q_j\cap Q_k\ne \mathrm{\varnothing }$, we have $h(x)<ch(y)$ 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 $d(x)=\mathrm{dist}(x,\partial D)$ and then put­ting, $${\rho }_D^{}(x_1,x_2)=\underset{\gamma }{inf}{\int }_{\gamma }\frac{ds}{d(\gamma (s))},$$ 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 $0<p<\mathrm{\infty }$. Us­ing this, Bañuelos ob­tains for $D$ a bounded Hölder of or­der 0 do­main so that $$\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{vol}(D_m{)}^{2/d}<\mathrm{\infty }.$$ This im­plies that $H(0)$ do­mains are reg­u­lar enough so that an ana­log of $\text{(3)}$ 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 $\phi :B(0,1)\to D$ maps the unit disc $B(0,1)$ of the com­plex plane con­form­ally onto $D$ with $\phi (0)=x$, then $$g_{\ast }^2(\phi )(\theta )=\frac{1}{\pi }{\int }_{B(0,1)}\log (\frac{1}{|z|})\frac{1-|z{|}^2}{|z-e^{i\theta }{|}^2}|{\phi }^{\prime }(z){|}^{2}dz$$ 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 $$\begin{array}{rl}\text{(4)}& E_x^h[{\tau }_D^{}]& =\frac{1}{h(x)}{\int }_D{G}_D(x,y)h(y)dy& =\frac{1}{h(x)}{\int }_{B(0,1)}\log (\frac{1}{|z|})h(\phi (z))|{\phi }^{\prime }(z){|}^{2}dxdy\\ & =\frac{1}{h(x)}{\int }_0^{2\pi }{g}_{\ast }^2(\phi )(\theta )d\mu (\theta ).\end{array}$$ Since $\mu ([0,2\pi ])=h(x)$ and $$g_{\ast }^2(\phi )(\theta )\le C{\int }_{B(0,1)}|{\phi }^{\prime }(z){|}^{2}dz\le C\mathrm{vol}(D),$$ it fol­lows that $$E_x^h[{\tau }_D^{}]\le C\mathrm{vol}(D),$$ 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 $$c\underset{\phi }{sup}{g}_{\ast }^2(\phi )(0)\le \underset{\phi }{sup}{\int }_{B(0,1)}K(z,1)K(z,-1)|{\phi }^{\prime }(z){|}^{2}dxdy\le C\underset{\phi }{sup}{g}_{\ast }^2(\phi )(0),$$ where the $sup$ is taken over all con­form­al map­pings $\phi :B(0,1)\to D$ with $\phi (0)=x$. But an­oth­er equi­val­ence holds for $K(z,1)K(z,-1)$. Namely, if $d(z,\mathrm{\Gamma })$ de­notes the hy­per­bol­ic dis­tance in $B(0,1)$ from $z$ to the geodes­ic $\tilde{\mathrm{\Gamma }}=[-1,1]$, then $$\frac{1}{4}K(z,1)K(z,-1)\le e^{-2d(z,\tilde{\mathrm{\Gamma }})}\le K(z,1)K(z,-1).$$ 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 $$c\underset{\mathrm{\Gamma }}{sup}{\int }_D{e}^{-2d_D^{}(z,\mathrm{\Gamma })}\le \underset{x,h}{sup}{E}_x^h[{\tau }_D^{}]\le C\underset{\mathrm{\Gamma }}{sup}{\int }_D{e}^{-2d_D^{}(z,\mathrm{\Gamma })}.$$ 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 $\mathrm{\Gamma }$ the hy­per­bol­ic geodes­ic con­nect­ing them, there are pos­it­ive con­stants $c$ and $C$ such that $$\frac{1}{4}{e}^{-C{d}_D^{}(z_Q^{},\mathrm{\Gamma })}\le E_{{\xi }_1}^{{\xi }_2}[T_Q]\le e^{-c{d}_D^{}(z_Q^{},\mathrm{\Gamma })}.$$ 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 $$P_Q=P_{{\xi }_1}^{{\xi }_2}({\tau }_{D\cap Q^c}^{}<{\tau }_D^{})\text{and}{P}_R=P_{{\xi }_1}^{{\xi }_2}({\tau }_{D\cap R^c}^{}<{\tau }_D^{})$$ and let­ting $\delta (D)$ be the area of the largest disc which can be in­scribed in $D$ yields $$\begin{array}{rl}& |{\mathrm{Cov}}_{{\xi }_1}^{{\xi }_2}(T_Q,T_R)|\le C{e}^{-c{\delta }_D^{}(z_Q^{},z_R^{})}\mathrm{vol}(Q)\mathrm{vol}(R)(P_Q+P_R)\text{and}\\ & {\mathrm{Var}}_{{\xi }_1}^{{\xi }_2}({\tau }_D^{})\le \delta (D)E_{{\xi }_1}^{{\xi }_2}[{\tau }_D^{}].\end{array}$$ 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, $$E_{{\xi }_1}^{{\xi }_2}[{\tau }_D^{}]\le c\text{and}{\mathrm{Var}}_{{\xi }_1}^{{\xi }_2}[{\tau }_D^{}]\le c/n.$$ 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 $\text{(2)}$ 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 $\mathrm{vol}(D)<\mathrm{\infty }$ in $d=2$, if $H^{+}(D)$ is the class of pos­it­ive har­mon­ic func­tions on $D$, then $$\begin{array}{}\text{(5)}& \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}\log P_x^h({\tau }_D^{}>t)=-{\lambda }_1\text{for }x\in D\text{ and }h\in H^{+}(D).\end{array}$$ This was ad­dressed in De Blassie [e13] where it was proved that $\text{(5)}$ 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 $L^{\mathrm{\infty }}$. To make this defin­i­tion pre­cise in the cur­rent set­ting, if ${\phi }_1$ is the first Di­rich­let ei­gen­func­tion for $\frac{1}{2}\mathrm{\Delta }$ on $D$, define a semig­roup on $L^2({\phi }_1^{2}dx)$ by $$P_t^{{\phi }_1}f(x)={\int }_D\frac{e^{{\lambda }_{1}t}p(t,x,y)}{{\phi }_1(x){\phi }_1(y)}f(y){\phi }_1^2(y)dy\text{for }f\in L^2({\phi }_1^{2}dx).$$ Then, the do­main $D$ is defined to be in­trins­ic­ally ul­tracon­tract­ive (IU) if there ex­ist con­stants $C_t$ such that $$|P_t^{{\phi }_1}f(x)|\le C_t\parallel f{\parallel }_{L^2({\phi }_1^{2}dx)}\text{for }t>0.$$ An im­port­ant con­sequence of IU is that for any $\epsilon >0$ there is a $t(\epsilon )$ such that $$\begin{array}{}\text{(6)}& (1-\epsilon )e^{-{\lambda }_{1}t}{\phi }_1(x){\phi }_1(y)\le p(t,x,y)\le (1+\epsilon )e^{-{\lambda }_{1}t}{\phi }_1(x){\phi }_1(y).\end{array}$$ Since, for any $h\in H^{+}(D)$, $$P_x^h({\tau }_D^{}\ge t)=\frac{1}{h(x)}{\int }_{D}p(t,x,y)h(y)dy\le 1,$$ it fol­lows eas­ily from $\text{(5)}$ that $$\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}\log P_x^h({\tau }_D^{}\ge t)=-{\lambda }_1,$$ giv­ing the Bañuelos ana­log of $\text{(2)}$ 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 $x\in D$: $$\underset{t\to \mathrm{\infty }}{lim}\frac{e^{{\lambda }_{1}t}p(t,x,y)}{{\phi }_1(x){\phi }_1(y)}=1\text{ uniformly in }y\in D.$$ This im­plies that the ana­log of $\text{(2)}$ 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 $$\underset{r\to 0}{lim}\underset{x\in {\mathbb{R}}^d}{sup}{\int }_{|x-y|<r}\frac{|V(y)|}{|x-y{|}^{d-2}}dy=0.$$ 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 $$\begin{array}{rl}\text{(7)}& \frac{1}{2}\mathrm{\Delta }u(x)+V(x)u(x)& =0\text{for }x\in D,u(x)& =f(x)\text{for }x\in \partial D.\end{array}$$ The Gauge The­or­em of Chung and Rao (see the art­icle of Ruth Wil­li­ams in this volume) says that either $E_x[e^{{\int }_0^{{\tau }_D^{}}V(B_s)ds}]\equiv \mathrm{\infty }$ 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 $\text{(7)}$ is giv­en by $$\begin{array}{r}u(x)=E_x[e^{{\int }_0^{{\tau }_D^{}}V(B_s)ds}f(B_{{\tau }_D^{}})].\end{array}$$

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, $$\begin{array}{}\text{(8)}& u(x)={\int }_{\partial D}{E}_x^y[e^{{\int }_0^{{\tau }_D^{}}V(B_s)ds}]f(y)P_x(B_{{\tau }_D^{}}\in dy).\end{array}$$ 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: $$\text{When is }{E}_x^y[e^{{\int }_0^{{\tau }_D^{}}V(B_s)ds}]<\mathrm{\infty }\text{?}$$ The quant­ity $$u(x,y)=E_x^y[e^{{\int }_0^{{\tau }_D^{}}V(B_s)ds}]$$ 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 $$E_x^y[e^{{\int }_0^{{\tau }_D^{}}V(B_s)ds}]\equiv \mathrm{\infty }$$ or there are pos­it­ive con­stants $c$ and $C$ such that $$\begin{array}{}\text{(9)}& c\le E_x^y[e^{{\int }_0^{{\tau }_D^{}}V(B_s)ds}]\le C\text{for all }x,y\in D\cup \partial D.\end{array}$$ 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 $$E_x^y[e^{{\int }_0^{{\tau }_D^{}}V(B_s)ds}]<\mathrm{\infty }$$ for one pair of points $x,y$, but $$E_z^w[e^{{\int }_0^{{\tau }_D^{}}V(B_s)ds}]=\mathrm{\infty }$$ 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 $\frac{1}{2}\mathrm{\Delta }$ on $D$, then there is a pos­it­ive con­stant $C$ such that $$\begin{array}{}\text{(10)}& \frac{G(x,z)G(z,y)}{G(x,y)}\le C(\frac{1}{|x-z{|}^{d-2}}+\frac{1}{|y-z{|}^{d-2}}).\end{array}$$ The left-hand side in $\text{(10)}$ 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 $\text{(10)}$ 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 $-\frac{1}{2}\mathrm{\Delta }$ and $-\frac{1}{2}\mathrm{\Delta }+V$. This lies close to the ori­gin­al mo­tiv­a­tion of Aizen­man and Si­mon [e2]. For ex­ample, sup­pose that for some $f\in C(\partial D)$, $$\begin{array}{rl}\frac{1}{2}\mathrm{\Delta }v(x)& =0\text{for }x\in D,\\ v(x)& =f(x)\text{for }x\in \partial D.\end{array}$$ and $$\begin{array}{rl}-\frac{1}{2}\mathrm{\Delta }u(x)+V(x)u(x)& =0\text{for }x\in D,\\ u(x)& =f(x)\text{for }x\in \partial D.\end{array}$$ 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 $\text{(4)}$ 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 $$-\frac{1}{2}\mathrm{\Delta }u(x)+V(x)u(x)=0.$$ Many oth­er sim­il­ar con­clu­sions fol­low in an equally easy man­ner. Us­ing the simple for­mula $$\begin{array}{}\text{(11)}& u(x,y)=\frac{G_V(x,y)}{G(x,y)},\end{array}$$ it fol­lows that $$cG(x,y)\le G_V(x,y)\le CG(x,y)\text{for }x,y\in D,$$ where $G_V$ is the Green func­tion for $-\frac{1}{2}\mathrm{\Delta }+V$. 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 $$cK(x,y)\le K_V(x,y)\le CK(x,y)\text{for }x,y\in D,$$ where $K$ and $K_V$ are the Mar­tin ker­nels for $\frac{1}{2}\mathrm{\Delta }$ and $-\frac{1}{2}\mathrm{\Delta }+V$, 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 $$c_{t}p(t,x,y)\le p_V(t,x,y)\le C_{t}p(t,x,y)\text{for }t>0\text{ and }x,y\in D,$$ where $p_V$ is the heat ker­nel for $-\frac{1}{2}\mathrm{\Delta }+V$. 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 $-\frac{1}{2}\mathrm{\Delta }+V$ 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, $(-\mathrm{\Delta }{)}^{\alpha }$ for $0<\alpha <2$, and po­ten­tials in the suit­ably mod­i­fied Kato class $K_{\alpha ,d}$, where $V\in K_{\alpha ,d}$ if $$\underset{r\to 0}{lim}\underset{\{x\in {\mathbb{R}}^d\}}{sup}{\int }_{|x-y|<r}\frac{|V(y)|}{|x-y{|}^{d-\alpha }}dy=0.$$ In this pa­per, Chen and Song [e31] de­duced the CGT on $C^{1,1}$ do­mains for the op­er­at­or $(-\mathrm{\Delta }{)}^{\alpha }$ 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 $\mathrm{\Delta }$. Their ap­proach was to split the po­ten­tial, writ­ing $V=V_1+V_2$ for $V_2\in {\mathbf{L}}^{\mathrm{\infty }}$ and $V_1$ with a small Kato norm, that is, with small $$\underset{x\in D}{sup}{\int }_D\frac{|V_1(y)|}{|x-y{|}^{d-\alpha }}dy.$$ Then, by a simple lemma of Khas­m­in­ski, they show that the Green func­tions $G_{V_1}^{\alpha }$ for $(-\mathrm{\Delta }{)}^{\alpha }+V_1$ and $G^{\alpha }$ for $(-\mathrm{\Delta }{)}^{\alpha }$ on $D$ sat­is­fy $G_{V_1}^{\alpha }\approx G^{\alpha }$, in the sense that there are pos­it­ive con­stants $c$ and $C$ such that $c{G}^{\alpha }\le G_{V_1}^{\alpha }\le C{G}^{\alpha }$. This equi­val­ence can then be used to prove that $(-\mathrm{\Delta }{)}^{\alpha }+V$ is IU. However, $(-\mathrm{\Delta }{)}^{\alpha }+V$ be­ing IU im­plies that $G^{\alpha }\approx G_V^{\alpha }$. Now, us­ing a for­mula ana­logue to $\text{(7)}$, the fi­nite­ness of the right-hand side fol­lows from the in­equal­ity $G_{V_1}^{\alpha }\le C{G}^{\alpha }$. 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 $$B_c=\{q:{\mathbb{R}}^d\to \mathbb{R}:\mathrm{supp}q\text{ is compact}\}\cap {\mathbf{L}}^{\mathrm{\infty }}.$$ The op­er­at­or $-\frac{1}{2}\mathrm{\Delta }+V$ is called sub­crit­ic­al if $$\text{for all }q\in B_c\text{ there exists }\epsilon >0\text{ such that }-\frac{1}{2}\mathrm{\Delta }+V+\epsilon q\ge 0.$$ This amounts to a strict pos­it­iv­ity of $-\frac{1}{2}\mathrm{\Delta }+V$.

For a sub­class of $K_d$ po­ten­tials which sat­is­fy a con­di­tion at $\mathrm{\infty }$, Zhao [e22] proved that sub­crit­ic­al­ity is equi­val­ent to $$u(x,y)=E_x^y[e^{{\int }_0^{\mathrm{\infty }}V(B_s)ds}]\text{ is bounded on }{\mathbb{R}}^d\times {\mathbb{R}}^d.$$ 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.