On Sullivan's construction of eigenfunctions via exit times of Brownian motion

TL;DR

This paper details Sullivan's method of constructing Laplacian eigenfunctions on Riemannian manifolds using Brownian motion exit times, providing a probabilistic approach to spectral analysis.

Contribution

It clarifies and elaborates on Sullivan's construction of eigenfunctions via exit times of Brownian motion on negatively curved manifolds.

Findings

Eigenfunctions can be represented as expected values involving exit times

The construction applies to manifolds with pinched negative curvature

Eigenfunctions correspond to complex eigenvalues with real part greater than the spectrum's supremum

Abstract

The purpose of this note is to give details for an argument of Sullivan to construct eigenfunctions of the Laplacian on a Riemannian manifold using exit times of Brownian motion \cite{sullivanpos}. Let $X$ be a complete, simply connected Riemannian manifold of pinched negative sectional curvature. Let $\lambda_1 = \lambda_1(X) < 0$ be the supremum of the spectrum of the Laplacian on $L^2(X)$, and let $D \subset X$ be a bounded domain in $X$ with smooth boundary. Let $(B_t)_{t \geq 0}$ be Brownian motion on $X$ and let $\tau = \tau_D$ be the first exit time of Brownian motion from $D$. For each $\lambda \in \mathbb{C}$ with $\hbox{Re } \ \lambda > \lambda_1$ and $x \in D$, we show that for any continuous function $\phi : \partial D \to \mathbb{C}$, the function $$ h(x) = \mathbb{E}_x(e^{-\lambda \tau} \phi(B_{\tau})) \ , \ x \in D, $$ is an eigenfunction of the Laplacian on $D$ with eigenvalue $\lambda$ and boundary value $\phi$.

Equations

$$h(x)={\mathbb{E}}_x(e^{-\lambda \tau }\varphi (B_{\tau })),x\in D,$$

$$h(x):={\mathbb{E}}_x(e^{-\lambda \tau }\varphi (B_{\tau }))$$

$$h(x)\to \varphi (\xi )$$

$$\Delta f=div(\nabla f)$$

$${\lambda }_1(X):=-\underset{\varphi }{\inf }\frac{{\int }_X|\nabla \varphi {|}^{2}dvol}{{\int }_X|\varphi {|}^{2}dvol},$$

$$e^{(s+t)\Delta }=e^{s\Delta }{e}^{t\Delta },s,t\ge 0.$$

$${\lambda }_1\le -(n-1{)}^2{a}^2/4,$$

$$\frac{\partial p}{\partial t}$$

$$p(t,x,.)$$

$${\int }_{X}p(t,x,y)f(y)dvol(y)\to f(x)ast\to 0$$

$$e^{t\Delta }f(x)={\int }_{X}p(t,x,y)f(y)dvol(y)$$

$$p(t,x,y)\le C{t}^{-n/2}\exp \left(c{\lambda }_{1}t-\frac{d(x,y{)}^2}{Dt}\right)$$

$$p(t,x,y)\le K{\left(1+\frac{d(x,y{)}^2}{t}\right)}^{1+\frac{n}{2}}\exp \left({\lambda }_{1}t-\frac{d(x,y{)}^2}{4t}\right)$$

$$J(x,v,r)\le K{e}^{hr}$$

$${\int }_{X-B(x,R)}p(t,x,y)dvol(y)\le \kappa e^{-\eta R^2/t}$$

$${\int }_{X-B(x,R)}p(t,x,y)dvol(y)$$

$$\le KC{t}^{-n/2}\sqrt{Dt}{\int }_{\frac{R}{\sqrt{Dt}}-\frac{\sqrt{Dt}h}{2}}^{\infty }{e}^{-s^2}ds$$

$${\int }_{X-B(x,R)}p(t,x,y)dvol(y)\le C_0{t}^{-(n-1)/2}{e}^{-R^2/(4Dt)}$$

$${\int }_A\mathbb{E}(\varphi |\mathcal{G})d\mathbb{P}={\int }_A\varphi d\mathbb{P}$$

$$\mathbb{E}(f(B_t)|{\mathcal{F}}_s)=\mathbb{E}(f(B_t)|B_s)$$

$$\mathbb{P}(B_0\in E)=\nu (E)$$

$$\mathbb{E}(f(B_{s+t})|{\mathcal{F}}_s)=(T(t)f)(B_s)$$

$${\int }_{\Omega }\mathbb{E}(f(B_t)|{\mathcal{F}}_0)d\mathbb{P}$$

$$\Rightarrow {\int }_{\Omega }f(B_t)d\mathbb{P}$$

$$\Rightarrow {\int }_{X}f(y)dP(t,x,y)$$

$$({\theta }_t{)}_{\ast }{\mathbb{P}}_x={\int }_X{\mathbb{P}}_{y}dP(t,x,y)$$

$$\underset{t\to 0}{\lim }\frac{1}{t}P(t,x,X-B(x,ϵ))=0$$

$${\int }_{X}p(t,x,y)dvol(y)\le 1$$

$${\int }_{X-B(x,r_t)}p(t,x,y)dvol(y)\le ϵ/M$$

$$\left|{\int }_{X}p(t,x,y)f(y)dvol(y)\right|$$

$$\le \frac{M}{2}\frac{ϵ}{M}+\frac{ϵ}{2}{\int }_{B(x,r_t)}p(t,x,y)dvol(y)$$

$$\le \frac{ϵ}{2}+\frac{ϵ}{2}=ϵ.$$

$${\int }_{X}p(t,x,y)dvol(y)=1$$

Full text

[TABLE]

is an eigenfunction of the Laplacian on $D$ with eigenvalue $\lambda$ and boundary value $\phi$.

1. 1 Introduction

2. 2 The heat kernel

3. 3 Construction of Brownian motion

4. 3.1 Markov processes and semigroups

5. 3.2 The heat semigroup and Brownian motion

6. 4 The infinitesimal generator of the heat semigroup

7. 5 The Dirichlet heat kernel of a precompact domain

8. 6 Exit times and eigenfunctions

1. Introduction

The purpose of this note is to give the details for a construction of Sullivan of eigenfunctions of the Laplacian on compact domains with smooth boundary using exit times for the Brownian motion which is just briefly sketched in [Sul87]. For simplicity, we restrict ourselves to manifolds with pinched negative curvature, though the arguments should still hold in a more general setting where the manifold is stochastically complete and the Brownian motion is transient.

Sullivan’s result, which is just stated in [Sul87] without any proof or reference unfortunately, is the following:

Theorem 1.1**.**

Let $X$ be a complete, simply connected Riemannian manifold of pinched negative sectional curvature, and let $(B_{t})_{t\geq 0}$ be Brownian motion on $X$. Let $\lambda_{1}(X)<0$ be the supremum of the Laplacian $\Delta$ on $L^{2}(X)$. Let $D\subset X$ be a precompact domain in $X$ with smooth boundary, and let $\tau$ be the first exit time from $D$ for Brownian motion. Then for any $\lambda\in\mathbb{C}$ with $\hbox{Re }\ \lambda>\lambda_{1}$, and for any continuous function $\phi:\partial D\to\mathbb{C}$, the function

[TABLE]

is $C^{\infty}$ on $D$ and is an eigenfunction of $\Delta$ on $D$ with eigenvalue $\lambda$ and boundary value $\phi$, meaning $\Delta h=\lambda h$ and

[TABLE]

as $x\in D\to\xi\in\partial D$.

For $\lambda=0$, this is just the well-known classical solution of the Dirichlet problem using Brownian motion. The above theorem for other values of $\lambda>\lambda_{1}$ must be well known to experts, but owing to the lack of a proof or reference in Sullivan, it seemed worthwhile to write down the details. The article is organized as follows: in section 2 we present basic facts about the heat kernel on a complete Riemannian manifold. In section 3 we describe the construction of Brownian motion from the heat semigroup. In section 4 we show that the infinitesimal generator of the heat semigroup restricted to $C^{\infty}_{c}(X)$ is given by the Laplacian. In section 5 we introduce the Dirichlet heat kernel of a bounded domain, while in section 6 we give the proof of the above theorem.

2. The heat kernel

Let $X$ be a complete, simply connected Riemannian manifold with pinched negative sectional curvature $-b^{2}\leq K\leq-a^{2}$. Let $\Delta$ be the Laplacian on $X$, acting on functions $f\in C^{\infty}(X)$ by

[TABLE]

Gaffney [Gaf54] showed that the densely defined operator $\Delta$ on $L^{2}(X)$ with domain $C^{\infty}_{c}(X)$ is essentially self-adjoint, hence it has a unique self-adjoint extension on $L^{2}(X)$, also denoted by $\Delta$. The domain of $\Delta$ is given by $f$ in $L^{2}(X)$ such that $\Delta f$ (in the sense of distributions) is in $L^{2}(X)$. Define $\lambda_{1}=\lambda_{1}(X)\leq 0$ by

[TABLE]

where the infimum is taken over all non-zero $\phi$ in $C^{\infty}_{c}(X)$. Then the spectrum of $\Delta$ on $L^{2}(X)$ is contained in $(-\infty,\lambda_{1}]$ and $\lambda_{1}$ is the supremum of the spectrum [Cha84]. The functional calculus for self-adjoint operators then allows us to define the heat semigroup $(e^{t\Delta})_{t\geq 0}$ as a semigroup of bounded operators on $L^{2}(X)$ satisfying

[TABLE]

Mckean [Mck70] showed that the upper bound on sectional curvature $K\leq-a^{2}$ implies that

[TABLE]

where $n$ is the dimension of $X$, in particular $\lambda_{1}<0$.

It is known that the action of the semigroup $e^{t\Delta}$ on $L^{2}(X)$ is given by integrating against a kernel $p(t,x,y)$ called the heat kernel [Dod83]. The heat kernel is a positive smooth function $p:(0,\infty)\times X\times X\to(0,\infty)$ satisfying

[TABLE]

where the second condition means

[TABLE]

for all bounded continuous functions $f$ (the heat kernel satisfies $\int_{X}p(t,x,y)dvol(y)\leq 1$ so the above integral is well-defined for $f$ bounded). The semigroup $e^{t\Delta}$ is given by

[TABLE]

for $f\in L^{2}(X)$. For the existence of the heat kernel, see [Cha84], [Dod83]. If $X$ has Ricci curvature bounded from below, which is the case with our hypothesis of sectional curvature bounded below, then the heat kernel is unique [Cha84].

Grigoryan [Gri94] gave upper bounds on the heat kernel for complete, non-compact manifolds in a general setting. In our case of a complete simply connected manifold of pinched negative curvature, these specialize to the following estimates:

There exist constants $c,C>0$ and $D>4$ such that for all $x,y\in X,t>0$, we have

[TABLE]

For long-time asymptotics, this can be improved: for all $x,y\in X,t>1$, we have

[TABLE]

Since $X$ has sectional curvature bounded below, the volume growth of $X$ is at most exponential, and there are constants $K,h>0$ such that the Jacobian $J(x,v,r)$ of the map $v\in T^{1}_{x}X\mapsto exp_{x}(rv)$ at $v$ satisfies a bound

[TABLE]

for all $x\in X,v\in T^{1}_{x}X,r>0$. Together with (1) above, this leads to the following lemma:

Lemma 2.1**.**

There are constants $\kappa,\eta>0$ such that for all $x\in X$, for $R>0$ and $t>0$ such that $R\geq(Dh)t$ we have

[TABLE]

Proof: Integrating in geodesic polar coordinates centered at $x$ and using the estimates (1), (3) gives

[TABLE]

Now $R\geq(Dh)t$ gives $\frac{R}{\sqrt{Dt}}-\frac{\sqrt{Dt}h}{2}\geq\frac{R}{2\sqrt{Dt}}$, then using a standard bound for the tail of a Gaussian integral gives, for some constant $C_{0}>0$,

[TABLE]

and the last expression above is bounded by $\kappa e^{-\eta R^{2}/t}$, choosing $\kappa$ large enough and $\eta$ small enough. $\diamond$

3. Construction of Brownian motion

We first briefly recall the correspondence between Markov processes and (certain) semigroups of bounded operators, and then explain how the heat semigroup $(e^{t\Delta})_{t\geq 0}$ can be used to construct Brownian motion on a manifold.

3.1. Markov processes and semigroups

Let $(X,d)$ be a locally compact, separable metric space equipped with its Borel sigma-algebra $\mathcal{B}(X)$. Let $B(X)$ denote the Banach space of bounded measurable functions on $X$ equipped with the supremum norm and $C_{0}(X)$ the closed subspace of continuous functions on $X$ vanishing at infinity.

We recall that conditional expectation is defined as follows: given a probability space $(\Omega,\mathcal{F},\mathbb{P})$, for any $\phi\in L^{1}(\Omega,\mathcal{F},\mathbb{P})$ and any sigma algebra $\mathcal{G}\subset\mathcal{F}$, the conditional expectation of $\phi$ given $\mathcal{G}$ is the unique $\mathcal{G}$-measurable random variable $\mathbb{E}(\phi|\mathcal{G})\in L^{1}(\Omega,\mathcal{G},\mathbb{P})$ such that

[TABLE]

for all $A\in\mathcal{G}$ (the existence of the conditional expectation follows from an application of the Radon-Nikodym theorem to the signed measure $A\in\mathcal{G}\mapsto\int_{A}\phi d\mathcal{P}$ on $\mathcal{G}$). For a random variable $Y$ on $\Omega$, we define the conditional expectation $\mathbb{E}(\phi|Y)$ to be the conditional expectation of $\phi$ given the sigma algebra generated by $Y$.

Given a probability measure $\nu$ on $X$, a Markov process on $X$ with initial distribution $\nu$ is a collection $(B_{t})_{t\geq 0}$ of $X$-valued random variables on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ (i.e. each $B_{t}$ is a measurable map $B_{t}:\Omega\to X$) such that for all bounded measurable functions $f$ on $X$ we have the Markov Property

[TABLE]

for all $t\geq s$ (where $\mathcal{F}_{s}\subset\mathcal{F}$ is the sub-sigma algebra generated by the maps $B_{u},0\leq u\leq s$), and such that $B_{0}$ has distribution $\nu$, i.e.

[TABLE]

for all Borel sets $E\subset X$. By a sample path of the process, we mean a path in $X$ of the form $t\mapsto B_{t}(\omega)$, for some $\omega\in\Omega$.

Let $(T(t))_{t\geq 0}$ be a semigroup of bounded operators on $B(X)$. We say that a Markov process $(B_{t})_{t\geq 0}$ on $X$ corresponds to the semigroup $(T(t))_{t\geq 0}$ if for all $f\in B(X)$ and $s,t\geq 0$ we have

[TABLE]

If the above condition holds, then using the Markov property one can show that the initial distribution $\nu$ together with the semigroup $(T(t))_{t\geq 0}$ determine the finite dimensional distributions of the process $(B_{t})_{t\geq 0}$, i.e. all the probabilities of the form $\mathbb{P}(B_{t_{1}}\in E_{1},\dots,B_{t_{n}}\in E_{n})$ are determined, for any finite sequence $0\leq t_{1}<\dots<t_{n}$ and any Borel sets $E_{1},\dots,E_{n}\in\mathcal{B}(X)$.

Conversely, given a semigroup satisfying certain properties, one can construct a Markov process which corresponds to the semigroup and whose sample paths have good regularity properties. The precise statement is the following:

Let $(T(t))_{t\geq 0}$ be a semigroup of bounded operators on $B(X)$ which satisfies the following properties:

(a) $||T(t)||\leq 1$ for all $t\geq 0$ (contraction).

(b) $T(t)f\geq 0$ for all $f\geq 0$ (positivity).

(c) For all $f\in C_{0}(X)$, we have $T(t)f\in C_{0}(X)$ for all $t\geq 0$ (Feller property).

(d) For all $f\in C_{0}(X)$, we have $||T(t)f-f||\to 0$ as $t\to 0$ (strongly continuous).

(e) $T(t)1=1$ for all $t\geq 0$ (where $1$ denotes the function constant equal to 1) (conservative).

A semigroup satisfying these properties is called a Feller semigroup.

A classical result from probability theory (see for e.g. [EK86], Theorem 2.7 of Chapter 4) asserts that given a Feller semigroup $(T(t))_{t\geq 0}$ on a locally compact, separable metric space $X$, for any probability measure $\nu$ on $X$ there is a Markov process $(B_{t})_{t\geq 0}$ with initial distribution $\nu$ which corresponds to the semigroup $(T(t))_{t\geq 0}$, such that for $\mathbb{P}$-almost every $\omega$ the sample path $t\mapsto B_{t}(\omega)$ is a cadlag path, i.e. the path is right-continuous with left-limits existing for all $t$.

Let $D_{X}[0,\infty)$ denote the space of cadlag paths on $X$, for $t\geq 0$ let $\pi_{t}:D_{X}([0,\infty))\to X$ be the map defined by $\pi_{t}(\gamma):=\gamma(t)$, and let $\mathcal{D}$ be the sigma-algebra on $D_{X}([0,\infty))$ generated by the maps $\pi_{t},t\geq 0$. Given $x\in X$, by the above theorem, we have a Markov process $(B_{t})_{t\geq 0}$ on some $(\Omega,\mathcal{F},\mathbb{P})$ with initial distribution $\delta_{x}$ (the Dirac mass at $x$) which corresponds to the semigroup such that almost all sample paths are cadlag.

This gives a map $\Phi:\Omega\to D_{X}[0,\infty)$ sending $\omega\in\Omega$ to the sample path $t\mapsto B_{t}(\omega)$ (the map is defined $\mathbb{P}$-a.e.), and we thus obtain a probability measure $\mathbb{P}_{x}$ on $(D_{X}[0,\infty),\mathcal{D})$ defined by $\mathbb{P}_{x}:=(\Phi)_{*}\mathbb{P}$. For each $t\geq 0$, we then obtain a probability measure $P(t,x,.)$ on $X$ defined by $P(t,x,.):=(\pi_{t})_{*}\mathbb{P}_{x}$.

Since $\pi_{t}\circ\Phi=B_{t}$, it is clear that the measure $P(t,x,.)$ on $X$ is just the distribution of $B_{t}$ for the process corresponding to the semigroup with initial distribution $\delta_{x}$. The measures $P(t,x,.)$ are called the transition probabilities associated to the semigroup $(T(t))_{t\geq 0}$.

Since the process $(B_{t})_{t\geq 0}$ corresponds to the semigroup $(T(t))_{t\geq 0}$, and $B_{0}$ has initial distribution $\delta_{x}$, in fact we have for any $f\in B(X)$ and $t\geq 0$, using (4) and the definition of conditional expectation,

[TABLE]

(where in the last line we used $(B_{t})_{*}\mathbb{P}=P(t,x,.)$). Thus the action of the semigroup $(T(t))_{t\geq 0}$ on functions is given by integrating against the transition probabilities.

For $t\geq 0$, let $\theta_{t}:D_{X}[0,\infty)\to D_{X}[0,\infty)$ be the shift by time $t$ map, defined by $\theta_{t}(\gamma)(s)=\gamma(t+s)$. Then it can be shown that for any $x\in X$ and $t\geq 0$, the measure $(\theta_{t})_{*}\mathbb{P}_{x}$ is given by a convex combination of the measures $\mathbb{P}_{y},y\in Y$, namely

[TABLE]

(meaning that both sides above agree when applied to any $E\in\mathcal{D}$).

Finally, if we assume that for any $x\in X$ and $\epsilon>0$, the transition probabilities satisfy

[TABLE]

then for any probability measure $\nu$ on $X$, there is a Markov process $(B_{t})_{t\geq 0}$ with initial distribution $\nu$, which corresponds to the semigroup, such that almost all sample paths are continuous ([EK86], Proposition 2.9 of Chapter 4). Thus with hypothesis (7) above, we obtain in this case for each $x\in X$ a probability measure $\mathbb{P}_{x}$ on $(C_{X}[0,\infty),\mathcal{C})$, where $C_{X}[0,\infty)$ is the space of continuous paths on $X$ and $\mathcal{C}$ is the sigma-algebra generated by the coordinate maps $\pi_{t}:C_{X}[0,\infty)\to X,t\geq 0$ (it can be shown that this sigma-algebra coincides with the Borel sigma-algebra of $C_{X}[0,\infty)$ when $C_{X}[0,\infty)$ is equipped with the topology of uniform convergence on compact sets). The same relation (6) holds for the measures $\mathbb{P}_{x}$ on $C_{X}[0,\infty)$.

3.2. The heat semigroup and Brownian motion

Now let $X$ be a complete, simply connected Riemannian manifold of pinched negative sectional curvature. We indicate briefly how the heat semigroup gives a Feller semigroup on $B(X)$, whose transition probabilities satisfy the hypothesis (7). We will obtain therefore for each $x\in X$, a Markov process $(B_{t})_{t\geq 0}$ with initial distribution $\delta_{x}$, which corresponds to the heat semigroup, such that almost all sample paths are continuous. This process is called the Brownian motion on $X$ started at $x$. We obtain also a probability measure $\mathbb{P}_{x}$ on the space $C_{X}[0,\infty)$ of continuous paths in $X$, called the Wiener measure on paths started at $x$.

The heat kernel on any complete Riemannian manifold $X$ satisfies

[TABLE]

for all $t>0,x\in X$ (see section 2, Chapter VIII [Cha84]), so it follows that $(e^{t\Delta}f)(x)=\int_{X}p(t,x,y)f(y)dvol(y)$ is well-defined for any $f\in B(X)$ and satisfies $||e^{t\Delta}f||_{\infty}\leq||f||_{\infty}$. Thus the heat semigroup defines a contraction semigroup on $B(X)$. Positivity of the semigroup is clear since the heat kernel is positive.

We check the Feller property (c). Given $f\in C_{0}(X)$ and $\epsilon>0$, choose a ball $B(x_{0},R)$ such that $|f(y)|\leq\epsilon/2$ for $y\in X-B(x_{0},R)$ and choose $M>2||f||_{\infty}$. Given $t>0$, it follows from Lemma 2.1 that there is a constant $r_{t}>0$ such that

[TABLE]

for all $x\in X$. Then for $x$ such that $d(x,x_{0})>R+r_{t}$, we have $B(x,r_{t})\subset X-B(x_{0},R)$, thus

[TABLE]

It follows that $e^{t\Delta}f\in C_{0}(X)$.

The condition that the heat semigroup be conservative, $e^{t\Delta}1=1$, is equivalent to the condition

[TABLE]

for all $x\in X,t>0$. This is also referred to as stochastic completeness of $X$. Yau showed that any complete Riemannian manifold with Ricci curvature bounded below is stochastically complete [Yau78]. This holds in our case since the sectional curvature of $X$ is bounded below.

For the strong continuity of the heat semigroup we use the fact that any $f\in C_{0}(X)$ is uniformly continuous. Given $\epsilon>0$ let $\delta>0$ be such that $|f(x)-f(y)|<\epsilon/2$ for $d(x,y)<\delta$. From Lemma 2.1 it follows that there is a $t_{0}>0$ such that for $0<t<t_{0}$ we have

[TABLE]

for all $x\in X$ (where $M$ is a constant such that $M>4||f||_{\infty}$). Then for any $x\in X$, for $0<t<t_{0}$ we have

[TABLE]

Thus $||e^{t\Delta}f-f||_{\infty}\leq\epsilon$ for $0<t<t_{0}$, so the heat semigroup is strongly continuous.

This establishes that the heat semigroup is a Feller semigroup. It remains to check that the transition probabilities $P(t,x,.)$ for the heat semigroup satisfy the condition (7). From (3.1), the transition probability $P(t,x,.)$ is a measure on $X$ satisfying

[TABLE]

for all continuous bounded functions $f$ on $X$. It follows that the measure $P(t,x,.)$ is given by $dP(t,x,y)=p(t,x,y)dvol(y)$. Now condition (7) becomes

[TABLE]

which follows immediately from Lemma 2.1. This finishes the construction of Brownian motion on $X$.

It is customary to write $\mathbb{E}_{x}(\phi)$ for the expectation of a function $\phi$ on the sample space $\Omega$ of a Brownian motion started at $x\in X$. We then have the following fundamental formula: for any $t>0$ and $x\in X$,

[TABLE]

for any $f\in B(X)$. In particular, for any Borel set $E\subset X$, letting $f=1_{E}$ gives

[TABLE]

so that $p(t,x,.)$ is the density of the distribution of $B_{t}$ for Brownian motion started at $x$.

Finally, it will be useful to note that the sample space $\Omega$ of Brownian motion on $X$ can always be taken to be $C_{X}[0,\infty)$ with the random variables $B_{t}$ given by the coordinate maps $\pi_{t}:C_{X}[0,\infty)\to X$. Indeed, given a Brownian motion $(B_{t})_{t\geq 0}$ defined on some probability space $(\Omega,\mathcal{F},\mathbb{Q})$, we can define the map $\Phi:\Omega\to C_{X}[0,\infty)$ sending $\omega\in\Omega$ to the sample path $t\mapsto B_{t}(\omega)$, and let $\mathbb{P}$ be the probability measure on $C_{X}[0,\infty)$ defined by $\mathbb{P}=(\Phi)_{*}\mathbb{Q}$. Then the $X$-valued process $(\pi_{t})_{t\geq 0}$ on the probability space $(C_{X}[0,\infty),\mathcal{C},\mathbb{P})$ is a Brownian motion whose sample paths have the same distribution as those of $(B_{t})_{t\geq 0}$. This realization of Brownian motion will be referred to as the canonical coordinate process.

4. The infinitesimal generator of the heat semigroup

For a Markov process $(B_{t})_{t\geq 0}$ on $X$ corresponding to a semigroup $(T(t))_{t\geq 0}$ on $B(X)$, the infinitesimal generator of the semigroup is the operator $A$ with domain $D(A)\subset B(X)$ defined by

[TABLE]

where the domain $D(A)$ is the set of $f\in B(X)$ for which the above limit exists in $(B(X),||.||_{\infty})$. It is well known that if the semigroup is strongly continuous on $C_{0}(X)$ then the domain $D(A)$ is dense in $C_{0}(X)$ and the operator $A$ is closed.

For the heat semigroup $(e^{t\Delta})_{t\geq 0}$, it is natural to expect that the infinitesimal generator should be the Laplacian $\Delta$ acting on a suitable space of functions. We show the following:

Proposition 4.1**.**

For any $\phi\in C^{\infty}_{c}(X)$,

[TABLE]

Thus $C^{\infty}_{c}(X)$ is contained in the domain of the infinitesimal generator of the heat semigroup, and the generator restricted to $C^{\infty}_{c}(X)$ equals the Laplacian $\Delta$.

Proof: Recall Green’s identity: for any $f,g\in C^{\infty}(X)$ and any precompact domain $D\subset X$ with smooth boundary, we have

[TABLE]

where $\frac{\partial f}{\partial n},\frac{\partial g}{\partial n}$ are the normal derivatives on $\partial D$ and $dA$ is the Riemannian volume measure of $\partial D$.

For $f\in C^{\infty}(X)$ and $\phi\in C^{\infty}_{c}(X)$, applying the above formula on a domain $D$ such that $supp(\phi)\subset D$ (so that $\phi=\frac{\partial\phi}{\partial n}=0$ on $\partial D$) gives

[TABLE]

Given $\phi\in C^{\infty}_{c}(X)$ and $x\in X$, let $u(t)=(e^{t\Delta}\phi)(x),t\geq 0$. Then for $t>0$, $u(t)=\int_{X}p(t,x,y)\phi(y)dvol(y)$, so for $t>0$ using (8) above we have

[TABLE]

Since $\lim_{t\to 0}u^{\prime}(t)$ exists, it follows that $u$ is $C^{1}$ on $[0,\infty)$, and we can write

[TABLE]

It follows that for $t>0$,

[TABLE]

using the strong continuity of the heat semigroup. $\diamond$

Corollary 4.2**.**

For $h\in B(X)$ with compact support,

[TABLE]

in the sense of distributions on $X$.

Proof: Let $\mathcal{D}(X)$ be the space of distributions on $X$ and let $(.,.)$ denote the pairing between $\mathcal{D}(X)$ and $C^{\infty}_{c}(X)$. Then for $h\in B(X)$ with compact support and $\phi\in C^{\infty}_{c}(X)$, the distribution $\Delta h$ is defined by $(\Delta h,\phi):=(h,\Delta\phi)$. We have, using self-adjointness of the operators $e^{t\Delta}$ on $L^{2}(X)$, that

[TABLE]

since $\frac{e^{t\Delta}-id}{t}\phi\to\Delta\phi$ with respect to $||.||_{\infty}$ by Proposition 4.1, and $h\in L^{1}(X)$. $\diamond$

5. The Dirichlet heat kernel of a precompact domain

Let $D\subset X$ be a precompact domain in $X$ with smooth boundary $\partial D$. Let $H^{1}(D)$ be the Sobolev space

[TABLE]

(where $\nabla f$ is understood in the sense of distributions), equipped with the Sobolev norm

[TABLE]

Let $H^{1}_{0}(D)\subset H^{1}(D)$ be the closure of $C^{\infty}_{c}(D)$ with respect to the Sobolev norm. The Laplacian $\Delta$ with domain $C^{\infty}_{c}(D)\subset L^{2}(D,dvol)$ turns out to be essentially self-adjoint on $L^{2}(D)$. Its closure is called the Dirichlet Laplacian on $D$ and is denoted by $\Delta_{D}$. The domain of the Dirichlet Laplacian is given by

[TABLE]

(where $\Delta f$ is understood in the sense of distributions).

In this case $L^{2}(D)$ has an orthonormal basis of smooth eigenfunctions $\{\phi_{j}\}_{j\geq 1}$ of the Dirichlet Laplacian ([Cha84]) with eigenvalues

[TABLE]

(the multiplicity of the eigenvalue $\lambda_{1}(D)$ is one). The Dirichlet heat semigroup $(e^{t\Delta_{D}})_{t\geq 0}$ can be defined using the functional calculus for self-adjoint operators, and it admits an integral kernel $p_{D}(t,x,y)$, called the Dirichlet heat kernel. The Dirichlet heat kernel has the following expansion with respect to the basis $\{\phi_{j}\}$:

[TABLE]

(the series converges absolutely and uniformly on $\overline{D}$ for all $t>0$).

The Dirichlet heat kernel can also be obtained as the smallest positive fundamental solution of the heat equation in $D$ with Dirichlet boundary conditions. The kernel $p_{D}:(0,\infty)\times\overline{D}\times\overline{D}$ is positive, continuous, smooth on $(0,\infty)\times D\times D$, and satisfies

[TABLE]

(where the second condition above means $\int_{D}p_{D}(t,x,y)f(y)dvol(y)\to f(x)$ as $t\to 0$ for all continuous bounded functions $f$ on $D$).

The Dirichlet heat kernel $p_{D}(t,x,y)$ can be obtained from the heat kernel $p(t,x,y)$ as follows (see [Cha84], Chapter VII):

As shown in [Cha84], Chapter VII, for each $x\in\overline{D}$, there is a solution to the heat equation which is a continuous function $g(.,x,.):[0,\infty)\times\overline{D}\to\mathbb{R}$ such that $g$ is $C^{\infty}$ on $(0,\infty)\times D$, satisfying

[TABLE]

The Dirichlet heat kernel is then given by

[TABLE]

To proceed further, we will need the following well-known parabolic maximum principle for the heat equation (see [Cha84], section VIII.1):

For $T>0$, let $D_{T}=(0,T)\times D$, and define the parabolic boundary $\partial_{P}D_{T}$ of $D_{T}$ by $\partial_{P}D_{T}:=(\{0\}\times\overline{D})\cup([0,T]\times\partial D)$. Then we have:

Proposition 5.1**.**

(Parabolic maximum principle). Let $u$ be a continuous function on $\overline{D_{T}}$ which is a $C^{\infty}$ solution of the heat equation on $D_{T}$. Then

[TABLE]

and

[TABLE]

It follows from the parabolic maximum principle that the function $g$ defined above satisfies $g(t,x,y)\leq 0$ for all $t\geq 0,x,y\in\overline{D}$, and hence

[TABLE]

where $p(t,x,y)$ is as before the heat kernel of the whole manifold $X$. In particular,

[TABLE]

and the estimates (1), (2) also apply to $p_{D}(t,x,y)$ for $x,y\in\overline{D}$.

Moreover, away from the boundary of $D$ and for small times we have the following estimate for the function $g$ which will be useful:

Lemma 5.2**.**

There are constants $\alpha,\beta>0$, such that for any compact $K\subset D$, if $\delta>0$ denotes the distance from $K$ to the boundary of $D$, then for all $t\in(0,1),x\in K$ we have

[TABLE]

Proof: Given $x\in K$ and $t\in(0,1)$, since $g(0,x,y)=0$ for all $y\in\overline{D}$, the parabolic maximum principle gives

[TABLE]

and the required estimate now follows from the estimate (1) for the heat kernel after choosing $\alpha>0$ large enough and $\beta>0$ small enough and noting that $d(x,y)\geq\delta$ for $x\in K,y\in\partial D$. $\diamond$

6. Exit times and eigenfunctions

Let $(B_{t})_{t\geq 0}$ be a Brownian motion on $X$ started at $x\in X$, realized as the canonical coordinate process $(\pi_{t})_{t\geq 0}$ on $(C_{X}[0,\infty),\mathcal{C},\mathbb{P}_{x})$. For $t\geq 0$, we let $\theta_{t}:C_{X}[0,\infty)\to C_{X}[0,\infty)$ be the shift by time $t$ map as before. For convenience, we will denote $C_{X}[0,\infty)$ by $\Omega$ in what follows.

We say that the Brownian motion is transient if

[TABLE]

It is well-known (see for eg. [Gri99], section 5) that the Brownian motion is transient if and only if

[TABLE]

In our case we have the estimate (2), from which it is clear that the above integral converges, since $\lambda_{1}<0$. Thus the Brownian motion is transient, which means with probability one it leaves every compact set in $X$ eventually.

Given a precompact domain $D\subset X$, the exit time from $D$, $\tau=\tau_{D}:C_{X}[0,\infty)\to[0,\infty]$, is defined by

[TABLE]

Since the Brownian motion is transient, $\mathbb{P}_{x}(\tau<\infty)=1$. So the exit point from $D$, $\pi=\pi_{D}:\{\tau<\infty\}\subset C_{X}[0,\infty)\to X$, given by

[TABLE]

is defined almost everywhere. We note that if the starting point $x$ lies in $D$, then by continuity of the sample paths the exit point lies on $\partial D$ almost surely, i.e. $\mathbb{P}_{x}(\pi\in\partial D)=1$. The random variable $\pi$ is commonly written as $B_{\tau}$.

The following relation between the exit time $\tau$ and the Dirichlet heat kernel $p_{D}(t,x,y)$ is well-known: for any $t>0$ and $x\in D$, we have

[TABLE]

This leads to the following proposition:

Proposition 6.1**.**

For any $\lambda\in\mathbb{C}$ in the half-plane $\{\hbox{Re}\ \lambda>\hbox{Re}\ \lambda_{1}\}$, there is a constant $C_{\lambda}$ such that

[TABLE]

for all $x\in D$. Thus the complex measure $e^{-\lambda\tau}d\mathbb{P}_{x}$ has finite total variation for all $x\in D$.

Proof: Since $|e^{-\lambda s}|=e^{-(\hbox{Re }\lambda)s}$ for $s$ real and $\mathbb{P}_{x}$ is a probability measure, if $\hbox{Re }\lambda\geq 0$ we are done, so we may as well assume that $\lambda$ is real and $\lambda_{1}<\lambda<0$. Let $G$ be the monotone decreasing function $G(s)=\mathbb{P}_{x}(\tau\geq s)$. From the relation (11), using $p_{D}(t,x,y)\leq p(t,x,y)$ and the estimate (2) it follows that $G(s)\leq C_{1}e^{\lambda_{1}s}$ for some constant $C_{1}>0$ independent of $x$, so $e^{-\lambda s}G(s)\to 0$ as $s\to\infty$. We can then integrate by parts to obtain:

[TABLE]

$\diamond$

We observe that for any $t>0$ we have

[TABLE]

We will need the following estimate on the $\mathbb{P}_{x}$-measure of the set $\{\tau<t\}$ for $x$ in a compact in $D$ and for small times $t$:

Lemma 6.2**.**

There are constants $\gamma,\beta^{\prime}>0$, such that for any compact $K\subset D$, if $\delta>0$ denotes the distance from $K$ to the boundary of $D$, then for all $x\in K$ and $0<t<min(1,\delta/(Dh))$ we have

[TABLE]

Proof: Given $x\in K$ and $t\in(0,1)$, from (11) and using (9), Lemma 5.2 and Lemma 2.1 we have

[TABLE]

where $\beta^{\prime}=min(\eta,\beta)$ and $\gamma=\kappa+\alpha vol(D)$. $\diamond$

We now come to the proof of Theorem 1.1. We fix a $\lambda\in\mathbb{C}$ with $\hbox{Re}(\lambda)>\lambda_{1}$. For $x\in D$, we define the measure $\mu_{x,\lambda}$ on $\partial D$ by

[TABLE]

where $\pi$ is the exit point map defined previously. Since $\mathbb{P}_{x}(\pi\in\partial D)=1$ for $x\in D$, the measure $\mu_{x,\lambda}$ is supported on $\partial D$. By Proposition 6.1, $\mu_{x,\lambda}$ is a complex measure on $\partial D$ of finite total variation.

For the rest of this section, we fix a continuous function $\phi:\partial D\to\mathbb{C}$. We then define a function $h$ on $D$ by

[TABLE]

We wish to show that $h$ is an eigenfunction of $\Delta$ on $D$ with eigenvalue $\lambda$, and that $h(x)\to\phi(\xi)$ as $x\to\xi\in\partial D$.

It will be convenient to write the function $h$ as follows: we first extend $\phi$ to a function on all of $X$ such that $\phi$ is continuous with compact support. We then define a function $\Phi:\{\tau<\infty\}\subset C_{X}[0,\infty)\to\mathbb{C}$ by

[TABLE]

We will also write $\phi(B_{\tau})$ for the random variable $\Phi$. From the definition of $\mu_{x,\lambda}$, we can write $h$ as

[TABLE]

We note that if $x\in X-\overline{D}$, then $\mathbb{P}_{x}(\tau=0)=1$ and $\mathbb{P}_{x}(\pi=x)=1$, so if we define $h$ by equation (13) above, then $h(x)=\phi(x)$ for $x\in X-\overline{D}$. Thus we can regard $h$ as a bounded function on $X$ with compact support, defined by (13) for all $x\in X$.

Lemma 6.3**.**

For any compact $K\subset D$, we have

[TABLE]

uniformly on $K$ as $t\to 0$.

Proof: We first note that for $t>0$, $\pi\circ\theta_{t}=\pi$ on the set $\{t\leq\tau<\infty\}$, and hence $\Phi\circ\theta_{t}=\Phi$ on $\{t\leq\tau<\infty\}$. Together with the relations (12) and (6), this leads to the following, for $x\in K\subset D$:

[TABLE]

where $A(x,t)=\int_{\{\tau<t\}}e^{-\lambda(\tau-t)}\Phi d\mathbb{P}_{x}$ and $B(x,t)=\int_{\{\tau<t\}}e^{-\lambda\tau\circ\theta_{t}}\Phi\circ\theta_{t}d\mathbb{P}_{x}$. Thus

[TABLE]

so to finish the proof it suffices to show that $A(x,t)=o(t),B(x,t)=o(t)$ as $t\to 0$, uniformly in $x\in K$.

Let $\delta>0$ be the distance from $K$ to the boundary of $D$, and let $M>0$ be such that $|\phi|\leq M$ on $X$ (and so $|\Phi|\leq M$ on $\{\tau<\infty\}$). Then for $0<t<min(1,\delta/(Dh))$ and $x\in K$, using Lemma 6.2 we have

[TABLE]

We now estimate the term $B(x,t)$. For this it will be convenient to use Holder’s inequality. Since $\hbox{Re}(\lambda)>\lambda_{1}$, we can chose $p>1$ such that $\lambda^{\prime}:=p\lambda$ also satisfies $\hbox{Re}(\lambda^{\prime})>\lambda_{1}$. We can then estimate the $L^{p}$ norm of $e^{-\lambda\tau\circ\theta_{t}}$ with respect to $\mathbb{P}_{x}$ using the relation (6) and Proposition 6.1 as follows:

[TABLE]

(where we used $\mathbb{P}_{y}(\tau=0)=1$ for $y\in X-\overline{D}$ above). Letting $q\in(1,\infty)$ be such that $1/p+1/q=1$, we can estimate $|B(x,t)|$ using Holder’s inequality and Lemma 6.2:

[TABLE]

It now follows from (14) that

[TABLE]

uniformly on $K$ as $t\to 0$. $\diamond$

We can now show that $h$ is an eigenfunction of $\Delta$ with eigenvalue $\lambda$:

Proposition 6.4**.**

The function $h$ is $C^{\infty}$ on $D$ and satisfies

[TABLE]

on $D$.

Proof: Let $(.,.)$ denote the pairing between distributions on $D$ and functions in $C^{\infty}_{c}(D)$. Given by $\psi\in C^{\infty}_{c}(D)$, let $K=supp(\psi)\subset D$. Then it follows from Lemma 6.3 above that

[TABLE]

On the other hand, by Corollary 4.2, we have

[TABLE]

It follows that $\Delta h=\lambda h$ as distributions on $D$, and hence by elliptic regularity $h$ is $C^{\infty}$ on $D$ and $\Delta h=\lambda h$ as functions on $D$. $\diamond$

To complete the proof of Theorem 1.1, we will need the following lemmas:

Lemma 6.5**.**

Let $\xi\in\partial D$. Then for any $t>0$,

[TABLE]

and

[TABLE]

Proof: Since for $t>0$ fixed, $p_{D}(t,.,.)$ is a continuous function on $\overline{D}\times\overline{D}$ which vanishes for $(x,y)\in\partial D\times\overline{D}$, we have $p_{D}(t,x,y)\to 0$ uniformly in $y\in\overline{D}$ as $x\to\xi$, hence

[TABLE]

which proves (15) above. For (16), since $|e^{-\lambda\tau}|\leq 1$ for $\hbox{Re }\ \lambda\geq 0$, we may as well assume that $\lambda$ is real and $\lambda_{1}<\lambda<0$. For $x\in D$ we define the function

[TABLE]

then as before we have $G_{x}(s)\leq C_{1}e^{\lambda_{1}s}$ for some constant $C_{1}>0$ independent of $x$, so

[TABLE]

Now the first term on the right-hand side above tends to zero as $x\to\xi$ by (15), while for the second term we can use the dominated convergence theorem as follows: from estimate (2), we can find a constant $C_{2}>0$ such that for $x,y\in\overline{D}$ and $s\geq t$, we have $p(s,x,y)\leq C_{2}e^{\lambda_{1}s}$. This gives

[TABLE]

since $\lambda>\lambda_{1}$. Hence dominated convergence applies, and so

[TABLE]

since $p_{D}(s,x,y)\to 0$ as $x\to\xi$ for all $s\geq t,y\in D$. This proves (16). $\diamond$

For $x\in X$ and $\delta>0$, we define the exit time $\tau_{x,\delta}:\Omega\to[0,\infty]$ of Brownian motion from the ball $B(x,\delta)$ by

[TABLE]

Lemma 6.6**.**

For any $\delta>0$,

[TABLE]

uniformly in $x\in X$.

Proof: Given $x\in X$, let $B=B(x,\delta)$ and let $p_{B}(.,.,.)$ be the Dirichlet heat kernel of the ball $B$. As in section 5, we can write $p_{B}(.,x,.)=p(.,x,.)+g(.,x,.)$ where $g:[0,\infty)\times\overline{B}$ is continuous, is a solution of the heat equation on $(0,\infty)\times B$, and satisfies the boundary conditions $g(0,x,.)\equiv 0$, $g(t,x,y)=-p(t,x,y)$ for $t>0,y\in\partial B$. Lemma 5.2 applies in this situation to give constants $\alpha,\beta>0$ independent of $x$ such that

[TABLE]

Together with Lemma 2.1, this gives, for $0<t<\delta/(Dh)$,

[TABLE]

(where we have used the fact that $vol(B(x,\delta))\leq C_{\delta}$ for some constant independent of $x$, which holds since the sectional curvature of $X$ is bounded below). $\diamond$

We can now prove Theorem 1.1:

Proof of Theorem 1.1: It remains to show that $h(x)\to\phi(\xi)$ as $x\in D\to\xi\in\partial D$. Let $\xi\in\partial D$, and fix $\epsilon>0$. We choose $\delta>0$ such that $|\phi(y)-\phi(\xi)|\leq\epsilon$ for $y\in\partial D$ with $d(y,\xi)\leq\delta$, and fix a constant $M>||\phi||_{\infty}$. For any $t>0$, we can write the space $\Omega$ as

[TABLE]

where $\tau_{\xi,\delta}$ is the exit time from the ball $B(\xi,\delta)$ as defined previously. We then have, for $x\in D$,

[TABLE]

where

[TABLE]

We will show that by first choosing $t>0$ small enough, all the terms $C(x,t),D(x,t),E(x,t)$ are small for $x$ close enough to $\xi$.

Note that on the set $\{\tau<t,\tau<\tau_{\xi,\delta}\}$, for $x\in B(\xi,\delta)\cap D$,we have, $\mathbb{P}_{x}$-a.s., that $B_{\tau}\in B(\xi,\delta)\cap\partial D$, and hence $|\phi(B_{\tau})-\phi(\xi)|\leq\epsilon$ on this set $\mathbb{P}_{x}$-a.s. for $x\in B(\xi,\delta)\cap D$. Also on this set, since $\tau<t$, for $t>0$ small enough we have $|e^{-\lambda\tau}-1|\leq 2|\lambda|t$. We can then estimate, for $x\in B(\xi,\delta)\cap D$,

[TABLE]

for all $t\in(0,t_{1}]$, for some constants $C,t_{1}>0$ independent of $x$.

To estimate $D(x,t)$, we note that for $x\in B(\xi,\delta/2)$ we have $B(x,\delta/2)\subset B(\xi,\delta)$, and so $\mathbb{P}_{x}(\tau_{x,\delta/2}\leq\tau_{\xi,\delta})=1$ (since any path starting at $x$ must exit $B(x,\delta/2)$ before it exits $B(\xi,\delta)$), from which we get

[TABLE]

for $x\in B(\xi,\delta/2)$. Now from Lemma 6.6, it follows that there is $t_{2}>0$ such that for all $t\in(0,t_{2}]$ we have

[TABLE]

for all $x\in B(\xi,\delta/2)$. We may also assume choosing $t_{2}>0$ small enough that $t_{2}\leq t_{1}$, and $e^{|\lambda|t_{2}}\leq 2$. We now fix $t_{2}$. Then from Lemma 6.5, we can choose $\delta_{1}\in(0,\delta/2)$ such that

[TABLE]

for all $x\in B(\xi,\delta_{1})\cap D$. With these choices, we get for all $x\in B(\xi,\delta_{1})\cap D$ the estimate

[TABLE]

where we have used (6) and the way $t_{2},\delta_{1}$ were chosen.

Finally, we estimate $E(x,t_{2})$:

[TABLE]

From Lemma 6.5 it follows that there is a $\delta_{2}\in(0,\delta_{1})$ such that both terms on the right-hand side above are less than $\epsilon$ for $x\in B(\xi,\delta_{2})\cap D$, so that $E(x,t_{2})\leq 2\epsilon$ for $x\in B(\xi,\delta_{2})\cap D$.

Putting together all the estimates, we get, for $x\in B(\xi,\delta_{2})\cap D$,

[TABLE]

and so $h(x)\to\phi(\xi)$ as $x\to\xi$. $\diamond$