Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

Introduction

Let $M$ be a $d$-dimensional connected complete Riemannian manifold possibly with a convex boundary $\partial M$. Let $\rho$ be the Riemannian distance. Consider $L=\Delta+Z$ for the Laplace-Beltrami operator $\Delta$ and some $C^{1}$-vector field $Z$ such that the (reflecting) diffusion process generated by $L$ is non-explosive. Then the associated Markov semigroup $P_{t}$ is the (Neumann if $\partial M\neq\emptyset$) semigroup generated by $L$ on $M$. In particular, it is the case when the curvature of $L$ is bounded below; that is,

is equivalent to the curvature condition (1.1). Here, $\mathscr{P}(M)$ is the class of all probability measures on $M$; $W_{p}$ is the $L^{p}$-Warsserstein distance induced by $\rho$, i.e.,

where $\mathscr{C}(\mu_{1},\mu_{2})$ is the class of all couplings of $\mu_{1}$ and $\mu_{2}$; and for a Markov operator $P$ on $\mathscr{B}_{b}(M)$ (i.e. $P$ is a positivity-preserving linear operator with $P1=1$),

where $\nu(f):=\int_{M}f\text{\rm{d}}\nu$ for $f\in L^{1}(\nu)$. In some references, $\nu P$ is also denoted by $P^{*}\nu$. In the sequel we will use $P_{t}^{*}$ to stand for the adjoint operator of $P_{t}$ in $L^{2}(\mu)$ for the invariant probability measure $\mu$, hence adopt the notation $\nu P$ rather than $P^{*}\nu$ to avoid confusion. When the curvature is positive (i.e. $K>0$), (1.2) implies the $W_{p}$-exponential contraction of $P_{t}$ for $p\geq 1.$

In this paper, we aim to consider the case when (1.1) only holds for some negative constant $K,$ and to prove the exponential contraction

for some constants $c,\lambda>0$. It is crucial that the exponential rate $\lambda$ is independent of $p$. Due to the equivalence of (1.1) and (1.2), in the negative curvature case it is essential that $c>1$.

According to , even when $\text{\rm{Ric}}_{Z}$ is unbounded below, i.e. $\text{\rm{Ric}}_{Z}$ goes to $-\infty$ when $\rho_{o}:=\rho(o,\cdot)\rightarrow\infty$ for a fixed $o\in M$, there may exist the log-Sobolev inequality which implies the exponentially convergence of $P_{t}$ in entropy. This suggests that (1.3) may also hold for a class of diffusion semigroups with negative curvature.

for some constants $c,\lambda>0$, where $\delta_{x}$ is the Dirac measure at point $x$. Indeed, proved the $W_{1}$-exponential contraction with respect to a modified distance $f(|x-y|)$ in place of $|x-y|$ as constructed in for estimates of the spectral gap using the coupling by reflection. Under condition (1.4) the modified distance is comparable with the usual one so that (1.5) follows. As mentioned in that there is essential difficulty to prove (1.3) for $p>1$ even for this flat case.

In Luo and Wang the estimate (1.5) was extended as

for some constants $c,\lambda>0$. Comparing with (1.3) which is equivalent to

according to (see Proposition 3.1 below), (1.6) is less sharp for small $|x-y|$ and/or large $p$. It is open whether (1.4), or in the Riemannian setting that $\text{\rm{Ric}}_{Z}$ is uniformly positive outside a compact domain, implies (1.3) for some constants $c,\lambda>0$.

As in , we will consider the Warsserstein distances induced by Young functions in the class

For any $\Phi\in\mathscr{N}$ and a measure $\nu$ on $M$, consider the gauge norm in $L^{\Phi}(\nu):$

In particular, we have $\|f\|_{L^{\Phi_{p}}(\nu)}=\|f\|_{L^{p}(\nu)}$ for $\Phi_{p}(r):=r^{p},\ p\in(1,\infty)$. This is the reason why we do not take $\Phi_{p}(r)=\frac{1}{p}r^{p}$ in the characterization of Legendre conjugates. We extend the notion $\Phi_{p}$ to $p=1,\infty$ by letting $\Phi_{1}(r)=r,\Phi_{\infty}=\lim_{p\rightarrow\infty}\Phi_{p}$ and $\|f\|_{L^{\Phi_{p}}(\nu)}=\|f\|_{L^{p}(\nu)}$ for all $p\in[1,\infty].$ Now, let

In particular, $W_{\Phi_{p}}=W_{p}$ for $p\in[1,\infty].$ We aim to prove the exponential decay

when (1.1) only holds for a negative constant $K$, where $\Phi^{-1}$ is the inverse of $\Phi(\neq\Phi_{\infty})$ and we set $\Phi^{-1}_{\infty}(1)=1$ by convention.

To extend condition (1.4) to the Riemannian setting, consider the index

where $\rho$ is the Riemannian distance, $\mathscr{R}$ is the curvature tensor; $\gamma:[0,\rho(x,y)]\rightarrow M$ is the minimal geodesic from $x$ to $y$ with unit speed; $\{J_{i}\}_{i=1}^{d-1}$ are Jacobi fields along $\gamma$ such that

holds for the parallel transform $P_{x,y}:T_{x}M\rightarrow T_{y}M$ along the geodesic $\gamma$, and $\{\dot{\gamma}(s),J_{i}(s):1\leq i\leq d-1\}$ ($s=0,\rho(x,y)$) is an orthonormal basis of the tangent space (at points $x$ and $y$, respectively).

Note that when $(x,y)\in{\rm Cut}(M)$, i.e. $x$ is in the cut-locus of $y$, the minimal geodesic may be not unique. As a convention in the literature, all conditions on the index $I$ are given outside ${\rm Cut}(M)$. We now extend condition (1.4) to the non-flat case as follows: for some constants $K_{1},K_{2}>0$,

In the flat case we have $I(x,y)=0$ and $\rho(x,y)=|x-y|$, so that this condition reduces back to (1.4). Moreover, the curvature condition (1.1) is equivalent to

so that (1.8) implies $\text{\rm{Ric}}_{Z}\geq-(K_{1}+K_{2}).$

In the next section, we state our main results and present examples. With condition (1.8) we first extend the main results of to the present Riemannian setting and give the exponential convergence of $P_{t}$ in $W_{2}$. Under the ultracontractivity and condition (1.1) for some $K<0$, our the second result ensures the desired inequality (1.7). Finally, we extend these results to SDEs with multiplicative noise by using explicit conditions on the coefficients. To prove these results, we make some preparations in Section 3. Complete proofs of the main results are addressed in Sections 4-6 respectively.

Main Results and examples

We first consider the Riemannian setting, then extend to SDEs with multiplicative noise by using explicit conditions on the coefficients instead of the less explicit curvature condition.

We start with condition (1.8). Besides the extension of (1.6), this condition also implies the hypercontractivity and the exponential convergence in $W_{2}$ for the semigroup $P_{t}$. For a measure $\mu$ and constants $p,q\geq 1$, let $\|\cdot\|_{L^{p}(\mu)\rightarrow L^{q}(\mu)}$ stand for the operator norm form $L^{p}(\mu)$ to $L^{q}(\mu)$. Recall that $P_{t}$ is called hypercontractive if it has a unique invariant probability measure $\mu$ and $\|P_{t}\|_{L^{2}(\mu)\rightarrow L^{4}(\mu)}=1$ holds for large $t>0$. By interpolation theorem, $\|P_{t}\|_{L^{2}(\mu)\rightarrow L^{4}(\mu)}=1$ can be replaced by $\|P_{t}\|_{L^{p}(\mu)\rightarrow L^{q}(\mu)}=1$ for some $\infty>q>p>1.$

Let $\eqref{EB'}$ hold for some constants $K_{1},K_{2}$ and $r_{0}>0$. Then:

There exist two constants $c,\lambda>0$ such that for any $\Phi\in\bar{\mathscr{N}}$ and $x,y\in M$,

$P_{t}$ has a unique invariant probability measure $\mu$ and the log-Sobolev inequality

holds for some constant $C>0$. Consequently, $P_{t}$ is hypercontractive.

There exist constants $c,\lambda>0$ such that

To illustrate this result, we present below a consequence with explicit curvature conditions in the spirit of . These conditions allow $\text{\rm{Ric}}_{Z}$ to be negative everywhere, for instance, when $-C_{1}\leq\text{\rm{Ric}}\leq-C_{2}$ and $C_{2}>-\nabla Z\geq\delta$ for some constants $C_{1}>C_{2}>\delta>0$. As indicated in Introduction that (1.8) implies $\text{\rm{Ric}}_{Z}\geq-(K_{1}+K_{2}),$ so in the following corollary we assume that $\text{\rm{Ric}}_{Z}$ is bounded below.

Assume that $\text{\rm{Ric}}_{Z}$ is bounded below. Let $\rho_{o}=\rho(o,\cdot)$ for a fixed point $o\in M$. If there exist constants $\sigma>0$ and $\delta>\sigma(1+\sqrt{2})\sqrt{d-1}$ such that

Next, we introduce sufficient conditions for (1.7) which allow $\text{\rm{Ric}}_{Z}$ to be negative. Due to technical reason, we will need the ultracontractivity of $P_{t}$, which is essentially stronger than the hypercontractivity. We call $P_{t}$ ultracontractive if $\|P_{t}\|_{L^{1}(\mu)\rightarrow L^{\infty}(\mu)}<\infty$ for all $t>0.$ The ultracontractivity implies that $P_{t}$ has a density $p_{t}(x,y)$ with respect to $\mu$ (called heat kernel) and

In references (see e.g. ), the ultracontractivity is also defined by $\|P_{t}\|_{L^{2}(\mu)\rightarrow L^{\infty}(\mu)}<\infty$ for $t>0$. When $P_{t}$ is symmetric in $L^{2}(\mu)$ we have

so that these two definitions are equivalent. However, when $P_{t}$ is non-symmetric, the former might be stronger than the latter. The appearance of the ultracontractivity in our study is very nature: by Theorem 2.3(1) we already have (1.7) for $\Phi=\Phi_{1}$ (the weakest case), and by the ultracontractivity we are able to deduce the inequality from $\Phi_{1}$ to $\Phi_{\infty}$ (the strongest case). On the other hand, the result also indicates that (1.7) implies the hypercontractivity of $P_{t}$.

Assume that $\text{\rm{Ric}}_{Z}$ is bounded below.

If $P_{t}$ is ultracontractive, then there exist constants $c,\lambda>0$ such that for any $\Phi\in\bar{\mathscr{N}}$,

Consequently, for any $p\in[1,\infty],t\geq 0$ and $\mu_{1},\mu_{2}\in\mathscr{P}(M)$,

On the other hand, if there exist constants $c,\lambda>0$ such that

then the log-Sobolev inequality $\eqref{LS}$ holds for $c=\frac{2c^{2}}{\lambda}$, so that $P_{t}$ is hypercontractive.

We note that in Theorem 2.3(1) we have $\|\rho\|_{L^{p}(\mu\times\mu)}<\infty$ for $p\in[1,\infty)$. Indeed, since $\text{\rm{Ric}}_{Z}$ is bounded below, by [23, Theorem 2.1] the ultracontractivity implies the super log-Sobolev inequality (3.3) below, so that due to Herbst we have $(\mu\times\mu)(\text{\rm{e}}^{r\rho^{2}})<\infty$ for all $r>0$ (see e.g. ). Therefore, $G_{\Phi}(t)<\infty$ for $t>0$ and $\Phi\in\mathscr{N}$ satisfying

In the symmetric case (i.e. $Z=\nabla V$ for some $V\in C^{2}(M)$), explicit sufficient conditions for the ultracontractivity have been introduced in by using the dimension-free Harnack inequality in the sense of . Together with a suitable exponential estimate on the diffusion process, this inequality implies $\|P_{t}\|_{L^{2}(\mu)\rightarrow L^{\infty}(\mu)}<\infty$ for $t>0$ and thus, $P_{t}$ is ultracontractive due to (2.5). The conditions can be formulated as

where $\Psi_{1},\Psi_{2}:(0,\infty)\rightarrow(0,\infty)$ are increasing functions such that

and for some constants $\theta\in(0,1/(1+\sqrt{2}))$ and $C>0,$

When Ric is bounded below, (2.11) as well as the second inequality in (2.9) hold for $\Psi_{2}$ being a large enough constant. In general, since $\int_{0}^{r}\Psi_{1}(s)\text{\rm{d}}s\geq 2\int_{0}^{r/2}\Psi_{1}(s)\text{\rm{d}}s$, (2.11) with $\theta=\frac{1}{4}<\frac{1}{1+\sqrt{2}}$ follows from

Since (2.5) fails for non-symmetric semigroups, we apply the inequality

due to the semigroup property. So, to ensure the ultracontractivity, we need an additional condition implying $\|P_{t}\|_{L^{1}(\mu)\rightarrow L^{2}(\mu)}<\infty$ (see Corollary 2.4(2) below).

To estimate $G_{\Phi}(t)$ in (2.6) using $\Psi_{1}$, we introduce

Obviously, the inverse function $\Lambda_{2}^{-1}$ exists on $(0,\infty)$, and since $\Lambda_{1}$ is increasing with $\Lambda_{1}(\infty)=\infty$, we have

Assume that $\eqref{4.3}$ and $\eqref{4.4}$ hold for some constants $\theta\in(0,1/(1+\sqrt{2}))$ and $C>0.$

If $P_{t}$ is symmetric, i.e. $Z=\nabla V$ for some $V\in C^{2}(M)$, then there exist constants $c,\lambda>0$ such that $\eqref{LL0}$ and $\eqref{LL1}$ hold for

If $P_{t}$ is non-symmetric but there exists continuous $h\in C(;[0,\infty))$ with $h(r)>0$ for $r>0$ such that $\int_{0}^{1}\frac{h(r)}{r}\text{\rm{d}}r<\infty$ and

then there exist constants $c,\lambda>0$ such that $\eqref{LL0}$ holds for

To conclude this part, we present a simple example to illustrate Corollary 2.4.

Let $M$ have non-positive sectional curvatures and a pole $o\in M$. Let $Z=Z_{0}-\delta\nabla\rho_{o}^{2+\varepsilon}$ outside a compact domain, where $\delta,\varepsilon>0$ are constants and $Z_{0}$ is a $C^{1}$ vector field with

Let $\Psi_{2}:(0,\infty)\rightarrow(0,\infty)$ be increasing such that

By (2.13), (2.14) and the Hessian comparison theorem, we see that (2.9), (2.10) and (2.12) hold with $\Psi_{1}(r)=c_{1}r^{\varepsilon}$ for some constant $c_{1}>0$. According to Corollary 2.4, there exist constants $c,\lambda>0$ such that for any $p\geq 1$,

2 SDEs with multiplicative noise

We intend to investigate the $W_{p}$-exponential contraction for $p\in[1,\infty)$. As mentioned in Introduction that existing results only apply to $p=1$ and $\sigma=I$, and as mentioned in that there is essential difficulty to prove (1.3) for $p>1$ even for $\sigma=I$. So, the present study is non-trivial.

Corresponding to that (1.1) implies (1.2) in the Riemannian setting, we have the following assertion.

Note that this result does apply to $p=\infty$ when $\sigma$ is non-constant. Next, as in the Riemannian case, we intend to prove the exponential contraction in $W_{p}$ when (2.16) only holds for some negative constant $K_{p}$. To this end, we need the SDE to be non-degenerate. The following result contains analogous assertions in Theorems 2.1 and 2.3, where the first assertion extends (1.5) to the multiplicative noise setting.

Assume that $\sigma\sigma^{*}\geq\lambda_{0}^{2}I$ for some constant $\lambda_{0}>0$.

If there exist constants $K_{1},K_{2},r_{0}>0$ such that $Z$ and $\sigma_{0}:=\sqrt{\sigma\sigma^{*}-\lambda_{0}^{2}I}$ satisfy

then there exist constants $c,\lambda>0$ such that

Let $P_{t}$ have a unique invariant probability measure $\mu$ such that the log-Sobolev inequality

holds for some constant $C>0$. If there exists a constant $K>0$ such that

Combining this with $\|\cdot\|_{HS}^{2}\leq d\|\cdot\|^{2}$, we see that (2.17) follows from the following more explicit condition:

Note that conditions in Theorem 2.5 and Theorem 2.6(1) are explicit. To illustrate Theorem 2.6(2)-(3), we present below sufficient conditions for the log-Sobolev inequality (2.18) and the ultracontractivity of $P_{t}$. For $a:=\sigma\sigma^{*}$ and $(g_{ij})_{1\leq i,j\leq d}:=a^{-1}$, we introduce the Christoffel symbols

for some constant $K_{0}$. If there exist constants $c_{1},c_{2}>0$ and $\delta>1$ such that

then $P_{t}$ has a unique invariant probability measure $\mu$ and there exists a constant $c>0$ such that

We now introduce a simple example to illustrate Theorem 2.6.

then (2.22) holds for some constant $K_{0}$. Moreover, it is easy to see that

holds for some constants $c_{1},c_{2}>0$. By Proposition 2.7 and Theorem 2.6(3), for any $p\in[1,\infty)$, there exist constants $\lambda,c>0$ such that

Preparations

This section includes some propositions which will be used to prove the results introduced in Section 2. We first recall a link between the Wasserstein distance and gradient estimates due to , then deduce the hyperboundedness and the exponential convergence in entropy from the log-Sobolev inequality for non-symmetric diffusion semigroups, and finally prove the exponential contraction in gradient for ultracontractive semigroups in a general framework including both diffusion and jump Markov semigroups.

Let $(E,\rho)$ be a geodesic Polish space, i.e. it is a Polish space and for any two different points $x,y\in E$, there exists a continuous curve $\gamma:\rightarrow E$ such that $\gamma_{0}=x,\gamma_{1}=y$ and $\rho(\gamma_{s},\gamma_{t})=|s-t|\rho(x,y)$ for $s,t\in.$ Then for any $f\in{\rm Lip}_{b}(E)$, the class of bounded Lipschitz functions on $E$, the length of gradient

is measurable. Moreover, let $P(x,\text{\rm{d}}y)$ be a Markov transition kernel and define the Markov operator

For any $\Phi\in\bar{\mathscr{N}}\setminus\{\Phi_{\infty}\}$, consider the Young norm induced by $\Phi$ with respect to $P$

and set $\|f\|_{L_{*}^{\Phi_{\infty}}(P)}(x)=P|f|(x).$ Then $\|\cdot\|_{L_{*}^{\Phi_{p}}}=\|\cdot\|_{L^{\Phi_{q}}}$ for $p\in[1,\infty],q=\frac{p}{p-1}.$ The following result follows from [16, Theorem 2.2, Remark 2 and Remark 3].

For any constant $C>0$ and $\Phi\in\bar{\mathscr{N}}$, the following statements are equivalent to each other:

$|\nabla Pf|\leq C\|\nabla f\|_{L_{*}^{\Phi}(P)}$ for $f\in{\rm Lip}_{b}(E).$

$W_{\Phi}(\delta_{x}P,\delta_{y}P)\leq C\rho(x,y),\ \ x,y\in E.$

When $\Phi=\Phi_{p}$ for $p\in[1,\infty]$, they are also equivalent to

$W_{p}(\mu_{1}P,\mu_{2}P)\leq CW_{p}(\mu_{1},\mu_{2}),\ \ \mu_{1},\mu_{2}\in\mathscr{P}(E).$

2 Hyperboundedness and exponential convergence in entropy

When $P_{t}$ is symmetric, it is well known that the hyperbounddeness, exponential convergence in entropy and the log-Sobolev inequality are equivalent each other, see and references within. In the non-symmetric case, the log-Sobolev inequality implies the former two properties if the generator $L$ and the symmetric part of the Dirichlet form $\mathscr{E}$ satisfy

for some constant $c_{0}>0$ and a reasonable class $\mathscr{D}$ of non-negative bounded functions, which is stable under $P_{t}$ and dense in $L^{p}_{+}(\mu):=\{f\in L^{p}(\mu):f\geq 0\}$ for any $p\geq 1$, see e.g. . In applications, it may be not easy to figure out the class $\mathscr{D}$ such that (3.2) holds. But in general this condition can be replaced by the following approximation formula Lemma 3.2 in the spirit of .

Now, consider the (Neumann) semigroup $P_{t}$ generated by $L:=\Delta+Z$ for a local bounded vector field $Z$ such that $P_{t}$ has a unique invariant probability measure $\mu$. Let

Then $(L,\mathscr{D}_{0})$ is dissipative (thus, closable) in $L^{1}(\mu)$ with closure $(L,\mathscr{D}_{1}(L))$ generating $P_{t}$ in $L^{1}(\mu)$, see e.g. and references within. Let

Let $f\in\mathscr{D}$ and $\psi\in C_{b}^{\infty}([{\rm ess}_{\mu}\inf f,\infty))$. There exists a sequence $\{f_{n}\}_{n\geq 1}\subset\mathscr{D}_{0}$ with $\inf f_{n}=\inf f$ such that $f_{n}\rightarrow f$ in $L^{m}(\mu)$ for any $m\geq 1$, $Lf_{n}\rightarrow Lf$ in $L^{1}(\mu)$, and

Since $f\in\mathscr{D}\subset\mathscr{D}_{1}(L)\cap L^{\infty}(\mu)$, there exists a uniformly bounded sequence $\{f_{n}\}_{n\geq 1}\subset\mathscr{D}_{0}$ such that $\inf f_{n}={\rm ess}_{\mu}\inf f$ and $f_{n}\rightarrow f,Lf_{n}\rightarrow Lf$ in $L^{1}(\mu)$. By the uniform boundedness, $f_{n}\rightarrow f$ in $L^{m}(\mu)$ for any $m\geq 1$. Since $\psi\in C_{b}^{\infty}([\inf f_{n},\infty))$,

This implies $\mu(Lg_{n})=0$ since $\mu$ is $P_{t}$-invariant. So, by the dominated convergence theorem,

Let $Z$ be a locally bounded vector field such that the (Neumann) semigroup $P_{t}$ generated by $L:=\Delta+Z$ has a unique invariant probability measure $\mu$.

holds for some $\beta\in C((0,\infty);(0,\infty))$, then for any constants $q>p\geq 1$ and $\gamma\in C((p,q);(0,\infty))$ such that $t:=\int_{p}^{q}\frac{\gamma(r)}{r}\text{\rm{d}}r<\infty,$ there holds

(1) According to Lemma 3.2, for any $f\in\mathscr{D}$ and $p>1$, there exists $\{f_{n}\}_{n\geq 1}\subset\mathscr{D}_{0}$ such that $f_{n}\rightarrow f^{\frac{p}{2}}$ in $L^{m}(\mu)$ for all $m\geq 1$, and

Applying (3.3) to $f_{n}$ and using (3.5), we obtain

for $\gamma(p):=\frac{\beta(4c(p)(1-p^{-1}))}{pc(p)}.$ Noting that $\mathscr{D}$ is $P_{t}$-invariant (i.e. $P_{t}\mathscr{D}\subset\mathscr{D}$) and dense in $L_{+}^{p}(\mu)$ for any $p\geq 1$, the desired assertion follows from the proof of [13, Corollary 3.13].

(2) It suffices to prove for $g\in\mathscr{D}$ with $\inf g>0.$ Applying Lemma 3.2 to $f=P_{t}g$ and $\psi(s)=1+\log s$, and using (3.4), we obtain

This implies the desired exponential estimate. ∎

3 Exponential contraction in gradient

In this part, we consider a general framework including both diffusion and jump processes. Let $(E,\mathscr{F},\mu)$ be a separable complete probability space, and let $P_{t}$ be a Markov semigroup on $L^{2}(\mu)$ with $\mu$ as invariant probability measure. Let $(L,\mathscr{D}(L))$ be the generator of $P_{t}$ in $L^{2}(\mu)$. We assume that there exists an algebra $\mathscr{A}\subset\mathscr{D}(L)$ such that

$1\in\mathscr{A}$, $\mathscr{A}$ is dense in $L^{2}(\mu)$ and the algebra induced by

$\Gamma(f,g):=\frac{1}{2}(L(fg)-fLg-gLf)$ gives rise to a non-degenerate positive definite bilinear form on $\mathscr{D}\times\mathscr{D}$; i.e., for any $f\in\mathscr{D}$, $\Gamma(f,f)\geq 0$ and it equals to if and only if $f$ is constant.

In particular, when $P_{t}$ is the (Neumann) semigroup generated by $L:=\Delta+Z$ on $M$ with $\text{\rm{Ric}}_{Z}$ bounded below, the assumption holds for

is closable and the closure $(\mathscr{E},\mathscr{D}(\mathscr{E}))$ is a conservative symmetric Dirichlet form. Although $P_{t}$ is not associated to $(\mathscr{E},\mathscr{D}(\mathscr{E}))$ when it is non-symmetric, we have

If $\|P_{t}\|_{L^{1}(\mu)\rightarrow L^{\infty}(\mu)}<\infty,$ then $P_{t}$ has a heat kernel $p_{t}(x,y)$ with respect to $\mu$, i.e.

We consider the $``$gradient” length $|\nabla_{\Gamma}f|=\sqrt{\Gamma(f,f)}$ induced by $\Gamma$. Note that for jump processes the length is non-local and thus essentially different from the usual gradient length. As shown below that estimates of $|\nabla_{\Gamma}P_{t}|$ have a close link to functional inequalities of the associated Dirichlet form.

Assume that there exist $t_{1}>0$ and $\eta\in C([0,\infty);(0,\infty))$ such that

Then there exist constants $c,\lambda,t_{2}>0$ such that for any $q\geq 1$ and $\eta_{q}\in C([0,\infty);(0,\infty))$, the gradient estimate

for some constants $C,\lambda>0$. By the second inequality in (3.7), for any $t>0$ and $f\in\mathscr{D}$ we have

Integrating both sides over $[0,t]$ leads to

Taking $t=t_{1}$ and noting that $\mu$ is the invariant probability measure of $P_{t}$, we obtain

Since $\mathscr{D}(\mathscr{E})$ is the closure of $\mathscr{D}$ under the $\mathscr{E}_{1}$-norm, this inequality also holds for $f\in\mathscr{D}(\mathscr{E}).$ By condition (ii), the symmetric Dirichlet form is irreducible. So, according to [38, Corollary 1.2] the defective Poincaré inequality (3.11) implies the Poincaré inequality

for some constant $\lambda>0$. By (3.6) and that $\mathscr{D}$ is dense in $L^{2}(\mu)$, the Poincaré inequality is equivalent to

On the other hand, by the second inequality in (3.7), for any $t>0$ and $f\in\mathscr{D}$ we have

Using $P_{t}f-\mu(f)$ to replace $f$ and integrating with respect to $\mu$, we obtain

Combining this with (3.13) and (3.12) we arrive at

for some constant $c_{1}>0$; that is, (3.10) holds for $t>1.$ Finally, (3.7) implies (3.10) for $t\in.$

(b) Next, we intend to find out a constant $t_{0}\geq t_{1}$ such that

Indeed, by (3.13) and the first inequality in (3.7), we obtain

where $c_{0}:=\|P_{t_{1}}\|_{L^{1}(\mu)\rightarrow L^{\infty}(\mu)}.$ This implies the desired assertion for $t_{0}>0$ such that $c_{0}^{2}\text{\rm{e}}^{-\lambda t_{0}}\leq\frac{1}{2}$.

(c) Finally, combining (3.7), (3.14), (3.10) and (3.12), we obtain

for some constants $c_{1},c_{2},c_{3}>0$. Then (3.9) holds for $t_{2}=2t_{0}.$ ∎

Proof of Theorem 2.1

The proofs of the other two assertions are based on the log-Sobolev inequality and the log-Harnack inequality derived in and respectively for bounded below $\text{\rm{Ric}}_{Z}$.

(a) For two different points $x,y\in M$, let $P_{x,y}:T_{x}M\rightarrow T_{y}M$ be the parallel displacement along the minimal geodesic $\gamma:[0,\rho(x,y)]\rightarrow M$ from $x$ to $y$, and let $M_{x,y}:=P_{x,y}-2\langle\cdot,\dot{\gamma}_{0}\rangle\dot{\gamma}_{\rho(x,y)}:T_{x}M\rightarrow T_{y}M$ be the mirror reflection. Both maps are smooth in $(x,y)$ outside the cut-locus ${\rm Cut}(M)$. According to and , the appearance of the cut-locus and/or a convex boundary helps for the success of coupling, i.e. it makes the distance between two marginal processes smaller. So, for simplicity, we may and do assume that both the cut-locus and the boundary are empty, see [2, Section 3] or [33, Chapter 2] for details.

where $\text{\rm{d}}_{I}$ denotes the Itô differential introduced in on Riemannian manifolds, $B_{t}$ is the $d$-dimensional Brownian motion, and $u_{t}$ is the horizontal lift of $X_{t}$ to the frame bundle $O(M)$. Then $X_{t}$ is a diffusion process generated by $L$. To construct the coupling by reflection for short distance and parallel displacement for long distance, we introduce a cut-off function $h\in C^{1}([0,\infty))$ which is decreasing such that $h(r)=1$ for $r\leq r_{0},$ $h(r)=0$ for $r\geq r_{0}+1$, and $\sqrt{1-h^{2}}$ is also in $C^{1}$, see e.g. [40, (3.1)] for a concrete example. To construct the coupling in the above spirit, we split the noise into two parts, i.e. to replace $\text{\rm{d}}B_{t}$ by $h(\rho(X_{t},Y_{t}))\text{\rm{d}}B_{t}^{\prime}+\sqrt{1-h(\rho(X_{t},Y_{t}))^{2}}\text{\rm{d}}B_{t}^{\prime\prime}$ for two independent Brownian motions $B_{t}^{\prime}$ and $B_{t}^{\prime\prime}$, then make reflection for the $B_{t}^{\prime}$ part and parallel displacement for the $B_{t}^{\prime\prime}$ part. More precisely, let $(X_{t},Y_{t})$ solve the following SDE on $M\times M$ for $(X_{0},Y_{0})=(x,y)$:

Since the coefficients of the SDE are at least $C^{1}$ outside the diagonal $\{(z,z):z\in M\}$, it has a unique solution up to the coupling time

We then let $X_{t}=Y_{t}$ for $t\geq T$ as usual. By the second variational formula and the index lemma (see e.g. the proof of [34, Lemma 2.3] and [29, (2.4)]), the process $\rho_{t}:=\rho(X_{t},Y_{t})$ satisfies

for some one-dimensional Brownian motion $b_{t}$. Thus, by condition (1.8),

Since $h(\rho_{t})=0$ for $\rho_{t}\geq r_{0}+1$ while $\text{\rm{d}}\rho_{t}<0$ when $\rho_{t}\geq r_{0}+1,$ this implies

On the other hand, since $h(\rho_{t})=1$ for $\rho_{t}\leq r_{0}$, as observed in we have

for some constants $c,\lambda>0$. Indeed, let

which proves (2.1). Therefore, the proof of (1) is finished since the second inequality therein is a simple consequence of (2.1).

(b) According to the proofs of [34, Proposition 3.1 and Theorem 1.1], our conditions imply that $P_{t}$ is hyperbounded; that is, $\|P_{t}\|_{2\rightarrow 4}<\infty$ holds for some $t>0$. Since (1.8) implies $\text{\rm{Ric}}_{Z}\geq-(K_{1}+K_{2})$, by the hyperboundedness and [23, Theorem 2.1], we have the defective log-Sobolev inequality

for some constants $C_{1},C_{2}>0$. Since the symmetric Dirichlet form $\mathscr{E}(f,g):=\mu(\langle\nabla f,\nabla g\rangle)$ with domain $H^{1,2}(\mu)$ is irreducible, according to (see also ), the log-Sobolev inequality (3.4) holds for some constant $C>0$, so that (2) is proved.

(c) According to [25, Theorem 1.10] (see for the case without boundary), the log-Sobolev inequality implies the Talagrand inequality

Next, let $P_{t}^{*}$ be the adjoint of $P_{t}$ in $L^{2}(\mu)$. By Proposition 3.3 for $P_{t}^{*}$ in place of $P_{t}$, the log-Sobolev inequality implies

Moreover, according to [36, Theorem 1.1], the curvature condition $\text{\rm{Ric}}_{Z}\geq-(K_{1}+K_{2})=:-K$ is equivalent to the log-Harnack inequality

By [39, Proposition 1.4.4(3)], this implies

Combining (4.4), (4.5) and (4.6), we obtain

for some constant $c_{1}>0$. Noting that $\text{\rm{Ric}}_{Z}\geq-K$ implies $|\nabla P_{t}f|\leq\text{\rm{e}}^{Kt}P_{t}|\nabla f|$ (see e.g. ), by Proposition 3.1 we have

for some constants $c,\lambda>0$. Therefore, the proof of (3) is finished. ∎

Proof of Theorem 2.3 and Corollary 2.4

(1) Since $\text{\rm{Ric}}_{Z}\geq-K$ for some constant $K\geq 0$, we have (see e.g. )

Combining this with Proposition 3.4 for $q=1$ and noting that $P_{t}|\nabla f|$ is continuous, we obtain

for some constants $c_{0},\lambda,t_{0}>0$. Obviously, (3.1) implies

According to Proposition 3.1, this is equivalent to

Combining this with (5.1) and the semigroup property, we arrive at

This together with (5.1) implies (2.6) for some constants $c,\lambda>0.$ Moreover, (2.7) follows from (2.6) according to Proposition 3.1.

Then using the standard semigroup calculation of Bakry-Emery, this implies

Since $\lim_{t\rightarrow\infty}P_{t}g=\mu(g)$ for $g\in\mathscr{B}_{b}(M)$ due to the ergodicity, by letting $t\rightarrow\infty$ we prove the log-Sobolev inequality for (3.4) for $C=\frac{2c^{2}}{\lambda}.$ ∎

We first observe that the proof of [34, Theorem 4.2] works also for the non-symmetric case with $\nabla Z$ in place of $\text{\rm{Hess}}_{V}$, so that

Since in the symmetric case we have $\|P_{t}\|_{L^{1}(\mu)\rightarrow L^{\infty}(\mu)}\leq\|P_{t/2}\|_{L^{2}(\mu)\rightarrow L^{\infty}(\mu)}^{2}$, the first assertion follows immediately from Theorem 2.3.

by Theorem 2.3 and (5.2) it suffices to prove

for some constant $c^{\prime}>0.$ According to [23, Theorem 2.1], (5.2) implies the super log-Sobolev inequality (3.3) for

for some (possibly different) constant $c>0$. Then Proposition 3.3 with $p=1,q=2$ and $\gamma(r):=\frac{trh(r-1)}{(r-1)\int_{0}^{1}s^{-1}h(s)\text{\rm{d}}s}$ implies (5.3).

Proofs of Theorems 2.5-2.6 and Proposition 2.7

Let $X_{t}(x)$ solve (2.15) with initial point $x$. By Itô’s formula and condition (2.16) we obtain

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

Then the desired assertion follows from Proposition 3.1. ∎

where $B_{t}^{\prime}$ and $B_{t}^{\prime\prime}$ are independent $d$-dimensional Brownian motions. For any $x\neq y$, let $X_{t}$ solve this SDE with $X_{0}=x$, and let $Y_{t}$ solve the following coupled SDE with $Y_{0}=y$:

That is, under the flat metric we have made coupling by reflection for $B_{t}^{\prime\prime}$ and coupling by parallel displacement for $B_{t}^{\prime}$. Obviously, the coupled SDE has a unique solution up to the coupling time

We set $Y_{t}=X_{t}$ for $t\geq T_{x,y}$ as usual. Then by (2.17) and Itô’s formula, we obtain

By repeating the argument leading to (4.3), it is easy see that (6.3) and (6.4) imply

for some constants $c,\lambda>0$ independent of $x,y$. Therefore,

so that the first assertion follows from Proposition 3.1.

(2) According to [37, Theorem 1.1], $a\geq\alpha I$ and (2.19) imply the log-Harnack inequality

for some constants $c_{1},c_{2}>0$. Combining this with the log-Sobolev inequality, we prove the second assertion as in (c) in the proof of Theorem 2.1.

(3) According to the proof of Theorem 2.5, the condition (2.16) implies the gradient estimate (6.1). Next, by Proposition 3.4, the ultracontractivity and (6.1) imply

for some $c(p)>0$ and $\lambda>0$ independent of $p$. Then the proof if finished by Proposition 3.1. ∎

We will apply results in and . To this end, we introduce the Riemannian metric

and let $\Delta^{g},\nabla^{g},\text{\rm{Hess}}^{g}$ be the corresponding Laplacian, gradient and Hessian tensor respectively. Then $L=\Delta^{g}+Z$ for some $C^{1}$ vector field $Z$. We first verify the Bakry-Emery curvature condition (1.1) for some constant $K$. Using the Christoffel symbols, the intrinsic Hessian tensor induced by $g$ is formulated as

Thus, by Bochner-Weitzenböck formula and (2.22), at point $x$ there holds

for some constant $K_{1}$. Then (1.1) hold for some constant $K$.

Next, (2.23) implies that $P_{t}$ has a unique invariant probability measure $\mu$ such that $\mu(\text{\rm{e}}^{c_{2}|\cdot|^{2}})<\infty$ for some $c_{2}>\frac{K}{2\alpha}$. By our assumption on $a$, the Riemannian distance $\rho$ induced by the metric $g$ is equivalent to the Euclidian metric:

Then we may repeat the proof of [23, Corollary 2.5] with $\gamma(r)=c_{2}r^{\delta}$ and $\rho=|\cdot|$ to prove

for some constant $c_{3}>0.$ Combining this with the curvature condition (1.1), we obtain from [23, Theorem 2.1] for $p=2$ and $q=\infty$ that

holds for some constant $c_{4}>0$. Applying Proposition 3.3 below for $p=1,q=2$ and $\gamma(r)=c_{5}t(r-1)^{\frac{\delta-1}{2\delta}-1}$ for constant $c_{5}>0$ such that $t=\int_{1}^{2}\frac{\gamma(r)}{r}\text{\rm{d}}r$, we obtain

for some constant $c_{6}>0$. Combining this with (6.6) we arrive at

The author would like to thank Jian Wang for helpful comments.

References