Robin Heat Semigroup and HWI Inequality on Manifolds with Boundary

TL;DR

This paper develops a probabilistic approach to analyze the Robin heat semigroup on manifolds with boundary, establishing the HWI inequality and gradient estimates for noncompact manifolds with boundary.

Contribution

It introduces a probabilistic formula for the Robin heat semigroup using reflecting diffusion processes and derives the HWI inequality on manifolds with boundary, including nonconvex cases.

Findings

Probabilistic formula for Robin heat semigroup using reflecting diffusion.

HWI inequality established on manifolds with boundary.

Gradient estimates and Bismut's derivative formula for noncompact manifolds.

Abstract

Let $M$ be a complete connected Riemannian manifold with boundary $\pp M$, $Q$ a bounded continuous function on $\pp M$, and $L= \DD+Z$ for a $C^1$-vector field $Z$ on $M$. By using the reflecting diffusion process generated by $L$ and its local time on the boundary, a probabilistic formula is presented for the semigroup generated by $L$ on $M$ with Robin boundary condition $N,\nn f+Qf=0,$ where $N$ is the inward unit normal vector field of $\pp M$. As an application, the HWI inequality is established on manifolds with (nonconvex) boundary. In order to study this semigroup, Hsu's gradient estimate and the corresponding Bismut's derivative formula are established on a class of noncompact manifolds with boundary.

Equations

$$\text{Ric}-\nabla Z\ge -K$$

$${\partial }_{t}u=Lu,u(0,\cdot )=f,$$

$$u(t,x)=\mathbb{E}^{x}\big{[}f(X_{t})1_{\{t<\tau\}}\big{]}$$

$$u(t,x)={\mathbb{E}}^{x}f(X_t),$$

$$Nf:=⟨N,\nabla f⟩=-Qf\text{on}\partial M,$$

$$u(t,x)=P_{t}^{Q}f(x):=\mathbb{E}^{x}\big{\{}f(X_{t})\text{\rm{e}}^{\int_{0}^{t}Q(X_{s})\text{\rm{d}}l_{s}}\big{\}},\ \ \ t\geq 0,x\in M.$$

$$[0,r)\times \partial M\ni (s,x)\mapsto \exp [s{N}_x]\in {\partial }_{r}M$$

$${\mathcal{D}}_0:=\{f\in C_0^{\infty }(M):Nf+Qf=0\text{on}\partial M\}.$$

$$\mathcal{E}(f,g)=\mu (⟨f,g⟩)-{\mu }_{\partial }(Qfg),f,g\in {\mathcal{D}}_0.$$

$$\text{d}u(t-s,X_s)=\text{d}{M}_s+Nu(t-s,\cdot )(X_s)\text{d}{l}_s=\text{d}{M}_s-(Qu(t-s,\cdot ))(X_s)\text{d}{l}_s$$

$$|\nabla P_{t}f|\le {\text{e}}^{Kt}{P}_t|\nabla f|,f\in C_b^1(M).$$

$$\mu (f^2\log f^2)\le 2\sqrt{\mu (|\nabla f{|}^2)}{W}_2(f^2\mu ,\mu )+\frac{K}{2}{W}_2(f^2\mu ,\mu {)}^2,\mu (f^2)=1,$$

$$W_2(\nu ,\mu {)}^2:=\underset{\pi \in \mathcal{C}(\nu ,\mu )}{\inf }{\int }_{M\times M}\rho (x,y{)}^2\pi (\text{d}x,\text{d}y),$$

$$P_t(f^2\log f^2)\le (P_t{f}^2)\log P_t{f}^2+\frac{2({\text{e}}^{2Kt}-1)}{K}{P}_t|\nabla f{|}^2.$$

$$\mu (f^2\log f^2)\le \frac{2({\text{e}}^{2Kt}-1)}{K}\mu (|\nabla f{|}^2)+\mu ((P_t{f}^2)\log P_t{f}^2)).$$

$$\mu ((P_t{f}^2)\log P_t{f}^2)\le \frac{K{\text{e}}^{2Kt}}{2({\text{e}}^{2Kt}-1)}{W}_2(f^2\mu ,\mu {)}^2.$$

$$\mathbb{I}(X,X):=-⟨{\nabla }_{X}N,X⟩\ge -\sigma |X{|}^2,X\in T\partial M.$$

$$|\nabla P_{t}f|(x)\leq\text{\rm{e}}^{Kt}\mathbb{E}^{x}\big{\{}|\nabla f|(X_{t})\text{\rm{e}}^{\sigma l_{t}}\big{\}},\ \ \ x\in M,t\geq 0,f\in C_{b}^{1}(M),$$

$${\eta }_{\lambda }(s):=\underset{x\in M}{\sup }{\mathbb{E}}^x{\text{e}}^{\lambda l_s}<\infty ,s,\lambda \ge 0$$

$$\mu(f^{2}\log f^{2})\leq 4\bigg{(}\int_{0}^{t}\text{\rm{e}}^{2Ks}\eta_{2\sigma}(s)\text{\rm{d}}s\bigg{)}\mu(|\nabla f|^{2})+\frac{W_{2}(f^{2}\mu,\mu)^{2}}{4\int_{0}^{t}\text{\rm{e}}^{-2Ks}\eta_{2\sigma}(s)^{-1}\text{\rm{d}}s},\ \ \mu(f^{2})=1.$$

$${\delta }_r(Z):=\underset{{\partial }_{r}M}{\sup }⟨Z,\nabla {\rho }_{\partial M}{⟩}^{-},r>0.$$

$$0<r\leq\min\bigg{\{}{\rm i}_{\partial M},\ r_{0},\ \frac{1}{\sqrt{k}}\arcsin\bigg{(}\frac{\sqrt{k}}{\sqrt{k+\gamma^{2}}}\bigg{)}\bigg{\}},$$

$$\mu (f^2\log f^2)\le 2{\text{e}}^{2\sigma d/r}\sqrt{\mu (|\nabla f{|}^2)}{W}_2(f^2\mu ,\mu )+\frac{K_r{\text{e}}^{2\sigma d/r}}{2}{W}_2(f^2\mu ,\mu {)}^2,\mu (f^2)=1$$

$$\sup_{x\in M}\mathbb{E}^{x}\text{\rm{e}}^{\lambda l_{t}}\leq\exp\Big{[}\frac{\lambda dr}{2}+\Big{(}\frac{\lambda d}{r}+\lambda\delta_{r}(Z)+2\lambda^{2}\Big{)}t\Big{]},\ \ \ t\geq 0,\lambda\geq 0$$

$$0<r\leq\min\bigg{\{}{\rm i}_{\partial M},\ r_{0},\ \frac{1}{\sqrt{k}}\arcsin\bigg{(}\frac{\sqrt{k}}{\sqrt{k+\gamma^{2}}}\bigg{)}\bigg{\}}.$$

$$h(s)=\cos\big{(}\sqrt{k}\,s\big{)}-\frac{\gamma}{\sqrt{k}}\sin\big{(}\sqrt{k}\,s\big{)},\ \ \ s\geq 0.$$

$$h^{\prime \prime }+kh=0,h(0)=1,h^{\prime }(0)=-\gamma .$$

$$\Delta {\rho }_{\partial M}\ge \frac{(d-1)h^{\prime }}{h}({\rho }_{\partial M}),{\rho }_{\partial M}<{\mathrm{i}}_{\partial M}\wedge h^{(-1)}(0).$$

$$L{\rho }_{\partial M}\ge \frac{(d-1)h^{\prime }}{h}({\rho }_{\partial M})-{\delta }_r(Z),{\rho }_{\partial M}\le r.$$

$$\begin{array}{rl}& \alpha =(1-h(r){)}^{1-d}{\int }_0^r(h(s)-h(r){)}^{d-1}\text{d}s,\\ & \psi (s)=\frac{1}{\alpha }{\int }_0^s(h(t)-h(r){)}^{1-d}\text{d}t{\int }_{t\wedge r}^r(h(u)-h(r){)}^{d-1}\text{d}u,s\ge 0.\end{array}$$

Full text

AMS subject Classification: 60J60, 58G32.

Keywords: Gradient estimate, HWI inequality, local time.

1 Introduction

Let $M$ be a $d$-dimensional connected complete Riemannian manifold with boundary $\partial M$ and $L=\Delta+Z$ for some $C^{1}$-vector field $Z$ such that

[TABLE]

holds on $M$ for some constant $K\in\mathbb{R}$. This curvature condition is well known by Bakry and Emery [1].

Let $X_{t}$ be the reflecting diffusion process generated by $L$ on $M$, and let $l_{t}$ be its local time on the boundary $\partial M.$ Let $\tau$ be the first hitting time of $X_{t}$ to $\partial M$. It is well known that the following heat equation can be described by using the process $X_{t}$:

[TABLE]

where $f\in\mathscr{B}_{b}(M).$ With Dirichlet boundary condition $u|_{\partial M}=0$ the solution can be formulated as

[TABLE]

while under the Neumann boundary condition $Nu|_{\partial M}=0$ one has

[TABLE]

where $\mathbb{E}^{x}$ is the expectation taking for the process $X_{t}$ starting at $x.$ In this paper we shall provide the corresponding probability formula for the solution under the Robin boundary condition (cf. [4, page 102]):

[TABLE]

where $Q\in C_{b}(\partial M)$ and $N$ is the inward unit normal vector field on $\partial M$. It turns out that under a reasonable assumption the solution to (1.2) under condition (1.3) can be formulated by

[TABLE]

As soon as $P_{t}^{Q}$ is well defined, the semigroup property follows immediately from the Markov property of the reflecting diffusion process $X_{t}$. To ensure the boundedness of $P_{t}^{Q}$ under the uniform norm, it is natural to ask the local time $l_{t}$ to be exponentially integrable. According to calculations from [13] (see also the proof of Lemma 2.1 below), for this we shall need the following assumption.

(A)

*The boundary $\partial M$ has a bounded second fundamental form and a strictly positive injectivity radius, the sectional curvature of $M$ is bounded above, and there exists $r>0$ such that $Z$ is bounded on the $r$-neighborhood of $\partial M$. *

Let $\rho_{\partial M}$ be the Riemannian distance to the boundary. Then the $r$-neighborhood of $\partial M$ is $\partial_{r}M:=\{x\in M:\rho_{\partial M}(x)<r\},$ where $\rho_{\partial M}$ is the Riemannian distance to the boundary $\partial M$. Next, the injectivity radius ${\rm i}_{\partial M}$ of $\partial M$ is the largest number $r$ such that the exponential map

[TABLE]

is diffeomorphic. In particular, $\rho_{\partial M}$ is smooth on $\partial_{r}M$ for $r\leq{\rm i}_{\partial M}.$

Finally, to state our result, we introduce the following class of references functions:

[TABLE]

Theorem 1.1**.**

Assume (A) and $(\ref{C})$ hold and let $Q\in C_{b}(\partial M)$.

$(1)$* $\{P_{t}^{Q}\}_{t\geq 0}$ is a positivity-preserving strong Feller semigroup of bounded linear operators on $\mathscr{B}_{b}(M)$, whose generator is $L$ with domain containing all functions $f\in C^{2}_{0}(M)$ such that $Nf+Qf=0$ holds on $\partial M$.*

$(2)$* Let $Z=\nabla V$ for some $V\in C^{2}(M)$ with $\mu(\text{\rm{d}}x):=\text{\rm{e}}^{V(x)}\text{\rm{d}}x$ not necessarily finite. Then $\{P_{t}^{Q}\}_{t\geq 0}$ provides a bounded symmetric $C_{0}$-semigroup on $L^{2}(\mu).$ If in particular $Q\leq 0$, then $P_{t}^{Q}$ is sub-Markovian and the associated symmetric Dirichlet form $(\mathscr{E},\mathscr{D}(\mathscr{E}))$ is the closure of $(\mathscr{E},\mathscr{D}_{0})$ with*

[TABLE]

We note that a solution to the heat equation (1.2) under the Robin condition (1.3) can be represented by (1.4) provided it is bounded in $x\in M$. Indeed, by the boundary condition and the Itô formula, for fixed $t>0$,

[TABLE]

holds for some local martingale $M_{s}$ up to time $t$. So, $s\mapsto u(t-s,X_{s})\exp[\int_{0}^{s}Q(X_{r})\text{\rm{d}}l_{r}]$ is a local martingale as well. Since $u$ is bounded and $l_{t}$ is exponential integrable due to Lemma 2.1 below, it is indeed a martingale. Thus, (1.4) holds.

Since the local time $l_{t}$ is not absolutely continuous in $t$, the semigroup $P_{t}^{Q}$ is essentially different from the well developed Schrödinger semigroup. According to Theorem 1.1 below $P_{t}^{Q}$ is generated by $L$ under the boundary condition $Nf+Qf=0$, where $N$ is the inward unit normal vector field on $\partial M$. So, the formula (1.4) will be important in the study of this boundary value problem on $M$. In this paper we shall explain how can one apply this semigroup to the study of HWI inequality on manifolds with boundary. This inequality links three important quantities including the entropy, the energy and the Wasserstein distance (or the optimal transportation cost), and was found in [3, 2] on manifolds without boundary.

To study the HWI inequality, we consider the symmetric case that $Z=\nabla V$ for some $V\in C^{2}(M)$ such that $\mu(\text{\rm{d}}x)=\ \text{\rm{e}}^{V(x)}\text{\rm{d}}x$ is a probability measure on $M$, where $\text{\rm{d}}x$ is the Riemannian volume measure on $M$. Let $P_{t}$ be the semigroup of the reflecting diffusion process generated by $L$ on $M$, which is then symmetric in $L^{2}(\mu)$. When $\partial M$ is convex (1.1) implies the following gradient estimate (cf. [9, 12])

[TABLE]

Combining this estimate and an argument of [2], we can easily obtain the following HWI inequality:

[TABLE]

where $W_{2}$ is the $L^{2}$-Wasserstein distance induced by the Riemannian distance function $\rho$ on $M$. More precisely, for a probability measure $\nu$ on $M$ (note that we are using $\rho^{2}$ to replace $\frac{1}{2}\rho^{2}$ in [2])

[TABLE]

where $\mathscr{C}(\nu,\mu)$ is the class of all couplings of $\nu$ and $\mu.$

To see that $P_{t}^{Q}$ is important in the study of the HWI inequality on a nonconvex manifold, let us briefly introduce the main idea for the proof of (1.6) on a convex manifold using (1.5). Firstly, due to Bakry and Emery, (1.5) implies the semigroup log-Sobolev inequality

[TABLE]

Taking integration for both sides with respect to $\mu$ we arrive at

[TABLE]

On the other hand, according to [2, Proof of Lemma 4.2] the gradient estimate (1.5) implies (again note that the $W_{2}^{2}$ here is twice of the one in [2])

[TABLE]

Combining this with (1.8) and minimizing in $t>0$, one derives (1.6).

Now, what can we do for the nonconvex setting? According to [5], in this case the local time and the second fundamental form will be naturally involved in the upper bound of $|\nabla P_{t}f|$. Let the second fundamental form be bounded below by $-\sigma$ for some $\sigma\geq 0,$ i.e.

[TABLE]

Recall that $N$ is the inward unit normal vector field on $\partial M$. According to [5, Theorem 5.1], if $M$ is compact and $V=0$ then (1.1) and (1.10) imply

[TABLE]

where $X_{t}$ is the reflecting $L$-diffusion process and $l_{t}$ is its local time on $\partial M$. In this paper we shall prove (1.11) for $Z\neq 0$ on noncompact manifolds under assumption (A), see Proposition 2.2 below.

Since the local time is unbounded, we are not able to derive from (1.11) the semigroup log-Sobolev inequality like (1.7). But Theorem 1.1(2) enables us to derive a log-Sobolev inequality of type (1.8) using (1.11), from which we can prove the following HWI inequality (1.12).

Theorem 1.2**.**

Let $Z=\nabla V$ for some $V\in C^{2}(M)$ such that $\mu$ is a probability measure. Assume (A) and $(\ref{C})$. Let $\mathbb{I}\geq-\sigma$ for some $\sigma\in\mathbb{R}$. Then

[TABLE]

holds, and for any $t>0$,

[TABLE]

To derive an explicit HWI inequality, we shall estimate $\eta_{2\sigma}$ as in [13] by using the Itô formula for $\phi\circ\rho_{\partial M}(X_{t})$ with a specific choice of $\phi$ (see Lemma 2.1 below). From this we obtain the following consequence of Theorem 1.1 immediately. Let ${\rm Sect}_{M}$ be the sectional curvature of $M$, and let

[TABLE]

Corollary 1.3**.**

Let $Z=\nabla V$ for some $V\in C^{2}(M)$ such that $\mu$ is a probability measure. Assume (A) and $(\ref{C})$. Let $r_{0},\sigma,k,>0$ be such that $\delta_{r_{0}}(Z)<\infty,-\sigma\leq\mathbb{I}\leq\gamma$ and ${\rm Sect}_{M}\leq k$. For any

[TABLE]

and for $K_{r}:=K+\frac{\sigma d}{r}+\sigma\delta_{r}(Z)+4\sigma^{2},$ the HWI inequality

[TABLE]

holds.

As preparations, in the next section we shall confirm the exponential integrability of $l_{t}$ and establish (1.11) on noncompact manifolds. The above two theorems are then proved in Sections 3 and 4 respectively. To prove the strong Feller property of $P_{t}^{Q}$ and for further applications in the literature, the Bismut type formula for $P_{t}$ on manifolds with boundary is addressed in Appendix at the end of the paper.

2 Exponential estimate and Hsu’s gradient estimate

As explained in Section 1, to ensure that $P_{t}^{Q}$ is well defined, we first study the exponential integrability of the local time.

Lemma 2.1**.**

Let $r_{0}>0$ be such that $\delta_{r_{0}}(Z)<\infty$ and let $k,\gamma$ be in Corollary 1.3. Then

[TABLE]

holds for any

[TABLE]

Proof.

Let

[TABLE]

Then $h$ is the unique solution to the equation

[TABLE]

By the Laplacian comparison theorem for $\rho_{\partial M}$ (cf. [7, Theorem 0.3] or [14]),

[TABLE]

Thus,

[TABLE]

Now, let

[TABLE]

We have $\psi(0)=0,0\leq\psi^{\prime}\leq\psi^{\prime}(0)=1.$ Moreover, as observed in [13, Proof of Theorem 1.1],

[TABLE]

Combining this with (2.1) we obtain (note that $\psi^{\prime}(s)=0$ for $s\geq r$)

[TABLE]

On the other hand, since $\psi^{\prime}(0)=1$, by the Itô formula we have

[TABLE]

where $b_{t}$ is the one-dimensional Brownian motion. Then it follows from (2.2) and (2.3) that (note that $|\psi^{\prime}|\leq 1$)

[TABLE]

∎

This Lemma ensures the boundedness of $P_{t}^{Q}$ under the uniform norm. Next, we intend to prove (1.11) under assumption (A), which is known by [5] for compact $M$ and $Z=0.$

Proposition 2.2**.**

Assume that (A). Let $\kappa_{1},\kappa_{2}\in C_{b}(M)$ be such that

[TABLE]

hold on $M$ and $\partial M$ respectively. Then

[TABLE]

holds for all $f\in C_{b}^{1}(M),t>0,x\in M.$

We first provide a simple proof of (2.6) under a further condition that $|\nabla P_{\cdot}f|$ is bounded on $[0,T]\times M$ for any $T>0$, then drop this assumption by an approximation argument. Since this condition is trivial for compact $M$, our proof below is much shorter than that in [5].

Lemma 2.3**.**

Assume that $f\in C_{b}^{1}(M)$ such that $|\nabla P_{\cdot}f|$ is bounded on $[0,T]\times M$ for any $T>0$. Then $(\ref{Hsu})$ holds.

Proof.

For any $\varepsilon>0$, let

[TABLE]

By the Itô formula we have

[TABLE]

where $M_{s}$ is a local martingale. Combining this with (2.5) and (see [8, (1.14)])

[TABLE]

we obtain

[TABLE]

Since $\zeta_{s}$ is bounded on $[0,t],$ $\kappa_{1}$ and $\kappa_{2}$ are bounded, and by Lemma 2.1 $\mathbb{E}\text{\rm{e}}^{\lambda l_{t}}<\infty$ for all $\lambda>0$, this implies that

[TABLE]

is a submartingale for any $\varepsilon>0$. Letting $\varepsilon\downarrow 0$ we conclude that

[TABLE]

is a submartingale as well. This completes the proof.∎

By Lemma 2.3, to prove Proposition 2.2 it suffices to confirm the boundedness of $|\nabla P_{\cdot}f|$ on $[0,T]\times M$ for $f\in C_{b}^{1}(M).$ Below we first consider $f\in C_{0}^{\infty}(M)$ satisfying the Neumann boundary condition.

Lemma 2.4**.**

Assume (A). If $(\ref{C})$ holds then for any $T>0$ and $f\in C_{0}^{\infty}(M)$ such that $Nf|_{\partial M}=0$, $|\nabla P_{\cdot}f|$ is bounded on $[0,T]\times M.$

Proof.

We shall take a conformal change of metric as in [14] to make the boundary convex, so that the known estimates for the convex case can be applied. As explained on page 1436 in [14], under assumption (A) there exists $\phi\in C^{\infty}(M)$ and a constant $R>1$ such that $1\leq\phi\leq R,|\nabla\phi|\leq R,N\log\phi|_{\partial M}\geq\sigma,$ and $\nabla\phi=0$ outside $\partial_{r}M.$ Since $\mathbb{I}\geq-\sigma$, by [14, Lemma 2.1] $\partial M$ is convex under the new metric

[TABLE]

Let $\Delta^{\prime},\nabla^{\prime},\text{\rm{Ric}}^{\prime}$ be corresponding to the new metric. By [14, Lemma 2.2]

[TABLE]

As in [15] we shall now calculate the curvature tensor $\text{\rm{Ric}}^{\prime}-\nabla^{\prime}Z^{\prime}$ under the new metric. By [14, (9)], for any unit vector $U\in TM$, $U^{\prime}:=\phi U$ is unit under the new metric, and the corresponding Ricci curvature satisfies

[TABLE]

Noting that

[TABLE]

we have

[TABLE]

Combining this with (2.8), (1.1), $\|Z\|_{r}<\infty$ and the properties of $\phi$ mentioned above, we find a constant $K^{\prime}\geq 0$ such that

[TABLE]

For any $x,y\in M$, let $(X_{t}^{\prime},Y_{t}^{\prime})$ be the coupling by parallel displacement of the reflecting diffusion processes generated by $L^{\prime}$ with $(X_{0}^{\prime},Y_{0}^{\prime})=(x,y).$ Let $\rho^{\prime}$ be the Riemannian distance induced by $\langle\cdot,\cdot\rangle^{\prime}$. Since $(M,\langle\cdot,\cdot\rangle^{\prime})$ is convex, we have (see [12, (3.2)])

[TABLE]

Since $1\leq\phi\leq R$, we have $R^{-1}\rho\leq\rho^{\prime}\leq\rho$ so that

[TABLE]

To derive the gradient estimate of $P_{t}$, we shall make time changes

[TABLE]

Since $L^{\prime}=\phi^{2}L,$ we see that $X_{t}:=X_{\xi_{x}^{-1}(t)}^{\prime}$ and $Y_{t}:=Y^{\prime}_{\xi_{y}^{-1}(t)}$ are generated by $L$ with reflecting boundary. Again by $1\leq\phi\leq R$ we have

[TABLE]

Combining this with $|\nabla\phi|\leq R,1\leq\phi\leq R$ and (2.9) we arrive at

[TABLE]

Therefore,

[TABLE]

By (2.9) and $\xi_{y}^{-1}(t)\leq t$ we obtain

[TABLE]

Moreover, since $f\in C_{0}^{\infty}(M)$ with $Nf|_{\partial M}=0,$ it follows from the Itô formula and (2.10) that

[TABLE]

holds for some constant $c_{1}>0.$ Combining this with (2.11) and (2.12) we conclude that

[TABLE]

for some constant $c_{2}>0$.∎

Proof of Prposition 2.2. Let $f\in C_{b}^{1}(M)$. By Lemma 2.3 we only have to prove the boundedness of $|\nabla P_{\cdot}f|$ on $[0,T]\times M$ .

(a) Let $f\in C_{0}^{\infty}(M).$ In this case there exist a sequence of functions $\{f_{n}\}_{n\geq 1}\subset C_{0}^{\infty}(M)$ such that $Nf_{n}|_{\partial M}=0,f_{n}\to f$ uniformly as $n\to\infty$, and $\|\nabla f_{n}\|_{\infty}\leq 1+\|\nabla f\|_{\infty}$ holds for any $n\geq 1,$ see e.g. [11]. By Lemmas 2.3 and 2.4, (2.6) holds for $f_{n}$ in place of $f$ so that Lemma 2.1 implies

[TABLE]

for some constant $C>0.$ Letting first $n\to 0$ then $y\to x$, we conclude that $|\nabla P_{\cdot}f|$ is bounded on $[0,T]\times M.$

(b) Let $f\in C_{b}^{\infty}(M)$. Let $\{g_{n}\}_{n\geq 1}\subset C_{0}^{\infty}(M)$ be such that $0\leq g_{n}\leq 1,|\nabla g_{n}|\leq 2$ and $g_{n}\uparrow 1$ as $n\uparrow\infty$. By (a) and Lemma 2.3, we may apply (2.6) to $g_{n}f$ in place of $f$ such that Lemma 2.1 implies

[TABLE]

holds for some constant $C>0.$ By the same reason as in (a) we conclude that $|\nabla P_{\cdot}f|$ is bounded on $[0,T]\times M.$

(c) Finally, for $f\in C_{b}^{1}(M)$ there exist $\{f_{n}\}_{n\geq 1}\subset C_{b}^{\infty}(M)$ such that $f_{n}\to f$ uniformly as $n\to\infty$ and $\|\nabla f_{n}\|_{\infty}\leq\|\nabla f\|_{\infty}+1$ for any $n\geq 1.$ Therefore, the proof is complete by the same reason as in (a) and (b). ∎

3 Proof of Theorem 1.1

The boundedness of $P_{t}^{Q}$ under the uniform norm is ensured by Lemma 2.1. Since a bounded continuous function can be uniformly approximated by bounded smooth functions, due to Lemma 2.1 we may and do assume that $Q\in C_{b}^{\infty}(\partial M).$ To handle the integral $\int_{0}^{t}Q(X_{s})\text{\rm{d}}l_{s}$, we shall also need the upper bound of $L\rho_{\partial M}.$

Lemma 3.1**.**

Let $\mathbb{I}\geq-\sigma$ and $(\ref{C})$ hold. Then

[TABLE]

Proof.

Let $x\in M$ such that $\rho_{\partial M}(x)<{\rm i}_{\partial M}.$ Then there exist a unique $x_{0}\in\partial M$ and the minimal geodesic $x_{\cdot}:[0,\rho_{\partial M}(x)]\to M$ linking $\partial M$ and $x$. By (1.1) we have

[TABLE]

Let $h$ solve the equation

[TABLE]

By the Laplacian comparison theorem (see [6, Theorem 1]),

[TABLE]

Then the proof is completed by noting that

[TABLE]

∎

3.1 The strong Feller property

As explained after Theorem 1.1, when $M$ is compact the solution to (1.2) is bounded in $x\in M$, so that (1.4) holds. In particular, $P_{t}^{Q}f$ is differentiable for $f\in\mathscr{B}_{b}(M)$ and thus, $P_{t}^{Q}$ is strong Feller. When $M$ is noncompact, this argument does not apply due to the lack of boundedness of $u(t,\cdot).$ Below we provide a different proof for the strong Feller property.

a) We first prove the Feller property. Since by Lemma 2.1 for $f\in C_{b}(M)$ the function $P_{t}^{Q}f$ is bounded, it suffices to show that

[TABLE]

For any $y\in M$, let $(X_{s},Y_{s})$ be the coupling constructed in the proof of Lemma 2.4 via time changes. We shall first prove

[TABLE]

Using the notations in the proof of Lemma 2.4 and adopting (2.9) and (2.10), there exists a constant $c(t)>0$ such that for any $s\in[0,t]$

[TABLE]

By the continuity of the reflecting diffusion process we prove (3.2).

Next, to describe $\int_{0}^{t}Q(X_{s})\text{\rm{d}}l_{s}$ we shall apply the Itô formula to a proper reference function of $X_{s}$. To this end, we first extend $Q$ to a smooth function on $M$. By assumption (A), one may find a function $\tilde{Q}\in C^{\infty}(M)$ such that $\tilde{Q}|_{\partial M}=Q,N\tilde{Q}|_{\partial M}=0$ and $|\nabla\tilde{Q}|+|L\tilde{Q}|$ is bounded. This can be realized by using the polar coordinates

[TABLE]

for small enough $r>0$ such that $\rho_{\partial}$ is smooth on $\partial_{r}M$. From this one may take $\tilde{Q}(\theta,s)=Q(\theta)h(s)$ on $\partial_{r}M$ for some $h\in C^{\infty}([0,\infty)$ such that $h(0)=1,h^{\prime}(0)=0$ and $h(s)=0$ for $s\geq r,$ and let $\tilde{Q}=0$ outside $\partial_{r}M$. This $\tilde{Q}$ meets our requirements since $L\rho_{\partial}$ is bounded on $\partial_{r}M$ according to (2.1) and Lemma 3.1.

Let $\Phi\in C_{0}^{\infty}([0,\infty))$ be such that $0\leq\Phi\leq 1,\Phi(s)=1$ for $s\in[0,1]$ and $\Phi(s)=0$ for $s\geq 2.$ Let

[TABLE]

Then $0\leq\psi_{n}\leq 2n^{-1},\psi_{n}=\rho_{\partial M}$ for $\rho_{\partial M}\leq n^{-1},\psi_{n}$ is constant for $\rho_{\partial M}\geq 2n^{-1}$ and $|\nabla\psi_{n}|\leq 1.$ Moreover, $\psi_{n}\in C^{\infty}(M)$ for large $n$. Since $\nabla\psi_{n}=N$ and $N\tilde{Q}=0$ on $\partial M$, by the Itô formula we have

[TABLE]

where $M_{n}(t)$ is a martingale with

[TABLE]

Note that $L(\tilde{Q}\psi_{n})$ is bounded since so is $|\nabla\tilde{Q}|+|L\tilde{Q}|+1_{\partial_{r}M}|L\rho_{\partial M}|.$ Similarly, let $l_{s}^{y}$ be the local time of $Y_{s}$ on $\partial M$, we have

[TABLE]

for some martingale $M_{n}^{y}(t)$ with

[TABLE]

Combining these with (3.2) and using the dominated convergence theorem, we obtain

[TABLE]

and

[TABLE]

for some constant $c_{1}>0.$ Since by the construction of $\psi_{n}$, (3.3) and (3.4) we conclude that $M_{n}(t)$ and $M^{y}_{n}(t)$ are exponentially integrable uniformly in $y$ and

[TABLE]

holds for some constant $c_{2}>0$, it follows from (3.2) and (3.6) that

[TABLE]

Combining this with (3.5) we derive (3.1).

b) Let $f\in\mathscr{B}_{b}(M).$ By Remark A.1 in the Appendix, the Neumann semigroup is strong Feller. So, $f_{\varepsilon}:=P_{\varepsilon}f\in C_{b}(M)$ for any $\varepsilon>0.$ Combining this with the Feller property of $P_{t}^{Q}$, it suffices to prove

[TABLE]

Since $Q$ is bounded and $l_{s}$ is continuous in $s$ according to (2.4) and the continuity of $X_{s}$, Lemma 2.1 implies that

[TABLE]

Next, let $\{\mathscr{F}_{s}\}_{s\geq 0}$ be the natural filtration of $X_{s}$. By the Markov property we have

[TABLE]

Combining this with (3.8) we prove (3.7).

3.2 The generator

We first prove that

[TABLE]

Since a function in $C_{0}(M)$ can be uniformly approximated by functions in $C_{0}^{\infty}(M)$ satisfying the Neumann boundary condition, we may assume that $f\in C_{0}^{\infty}(M)$ with $Nf|_{\partial M}=0.$ By the Itô formula we have

[TABLE]

which goes to zero as $t\to 0$. Noting that

[TABLE]

by Lemma 2.1 for $\lambda=\|Q\|_{\infty}$ and letting first $t\to 0$ then $r\to 0$, we obtain (3.9).

Next, let $f\in C_{0}^{2}(M)$ satisfy the boundary condition $Nf+Qf=0.$ We intend to prove that

[TABLE]

By the Itô formula and the boundary condition, we have

[TABLE]

for some martingale $M_{t}$. This implies

[TABLE]

So,

[TABLE]

and hence (3.10) follows from (3.9) since $Lf\in C_{0}(M)$.

3.3 The symmetry and $C_{0}$ property

Let $Z=\nabla V$. Since by Lemma 2.1

[TABLE]

holds for some constant $c(t)>0$, the boundedness of $P_{t}^{Q}$ in $L^{2}(\mu)$ follows from the fact that $\mu$ is $P_{t}$-invariant. Moreover, since

[TABLE]

by Lemma 2.1 and the strong continuity of $P_{t}$ in $L^{2}(\mu)$, we conclude that $P_{t}^{Q}$ is strongly continuous in $L^{2}(\mu)$ as well. So, it remains to prove that for any $f,g\in C_{0}(M)$

[TABLE]

We shall prove (3.11) by using symmetric Schrödinger semigroups to approximate $P_{t}^{Q}$.

Let $\tilde{Q}$ and $\psi_{n}$ be constructed above. We have

[TABLE]

where

[TABLE]

which goes to zero uniformly in $x$ as $n\to\infty$ according to Lemma 2.1 and the properties of $\tilde{Q}$ and $\psi_{n}$. Let $P_{t}^{(n)}$ be the Schrödinger semigroup generated by

[TABLE]

Since $L(\tilde{Q}\psi_{n})$ is bounded, by the Feynman-Kac formula

[TABLE]

and $P_{t}^{(n)}$ is symmetric in $L^{2}(\mu)$. So,

[TABLE]

Obviously, $M_{n}(t)-(\tilde{Q}\psi_{n})(X_{t})\to 0$ a.s. as $n\to\infty$ and is exponentially integrable uniformly in $x$. So, by Lemma 2.1, (3.12), (3.13) and using the dominated convergence theorem we arrive at

[TABLE]

3.4 The Dirichlet form

Again let $Z=\nabla V$ for $V\in C^{2}(M)$. Let $Q\leq 0$ such that $\mathscr{E}\geq 0.$ Since by (1.3) and the integration by parts formula, we have

[TABLE]

the form $\mathscr{E},\mathscr{D}_{0})$ is closable. Moreover, as in §3.2 for $f\in\mathscr{D}_{0}$ one has $\frac{P_{t}^{Q}f-f}{t}\to Lf$ in $L^{2}(\mu)$ as $t\to 0$, it remains to show that $(\mathscr{E},\mathscr{D}_{0})$ is a pre-Dirichlet form in $L^{2}(\mu)$. Firstly, to understand that $(\mathscr{E},\mathscr{D}_{0})$ is well defined in $L^{2}(\mu)$