# Stochastic Analysis on Path Space over Time-Inhomogeneous Manifolds with
Boundary

Li-Juan Cheng 111Correspondence should be addressed to Li-Juan Cheng (E-mail:
<EMAIL_ADDRESS>
(School of Mathematical Sciences, Beijing Normal University,
Laboratory of Mathematics and Complex Systems, Ministry of Education,
Beijing 100875, The People’s Republic of China)
E-mail: chenglj@mail.bnu.edu.cn(L.J. Cheng)

###### Abstract

Let $L_{t}:=\Delta_{t}+Z_{t}$ for a $C^{1,1}$-vector field $Z$ on a
differential manifold $M$ with possible boundary $\partial M$, where
$\Delta_{t}$ is the Laplacian induced by a time dependent metric $g_{t}$
differentiable in $t\in[0,T_{c})$. We first introduce the damp gradient
operator, defined on the path space with reference measure $\mathbb{P}$, the
law of the (reflecting) diffusion process generated by $L_{t}$ on the base
manifold; then establish the integration by parts formula for underlying
directional derivatives and prove the log-Sobolev inequality for the
associated Dirichlet form, which is further applied to the free path spaces;
and finally, establish numbers of transportation-cost inequalities associated
to the uniform distance, which are equivalent to the curvature lower bound and
the convexity of the boundary.

Keywords: Metric flow, log-Sobolev inequality, integration by parts formula,
path space over manifolds with boundary, reflecting $L_{t}$-diffusion process

MSC(2010): 60J60, 58J65, 53C44

##  1 Introduction

Let $M$ be a $d$-dimensional differential manifold with possible boundary
$\partial M$ equipped with a complete Riemannian metric
$(g_{t})_{t\in[0,T_{c})}$, which is $C^{1}$ in $t$. For simplicity, we take
the notations: for $X,Y\in TM$,

$\displaystyle{\rm Ric}_{t}^{Z}(X,Y):={\rm
Ric}_{t}(X,Y)-\left<\nabla^{t}_{X}Z_{t},Y\right>_{t},\ \
\mathcal{G}_{t}(X,X):=\partial_{t}g_{t}(X,X)$
$\displaystyle\mathcal{R}_{t}^{Z}(X,Y):={\rm
Ric}_{t}^{Z}(X,Y)-\frac{1}{2}\mathcal{G}_{t}(X,Y),$

where ${\rm Ric}_{t}$ is the Ricci curvature tensor with respect to $g_{t}$,
$(Z_{t})_{t\in[0,T_{c})}$ is a $C^{1}$-family of vector fields, and
$\left<\cdot,\cdot\right>_{t}:=g_{t}(\cdot,\cdot)$. Define the second
fundamental form of the boundary by

$\mathbb{I}_{t}(X,Y)=-\left<\nabla^{t}_{X}N_{t},Y\right>_{t},\ \ X,Y\in
T\partial M,$

where $N_{t}$ is the inward unit normal vector field of the boundary
associated with $g_{t}$; $T\partial M$ is the tangent space of $\partial M$.
Consider the elliptic operator $L_{t}:=\Delta_{t}+Z_{t}$. Let $X_{t}$ be the
reflecting inhomogeneous diffusion process generated by $L_{t}$ (called
reflecting $L_{t}$-diffusion process). Assume that $X_{t}$ is non-explosive
before $T_{c}$. In this case, for any $0\leq S<T<T_{c}$, the distribution
$\Pi^{S,T}$ of $X_{[S,T]}:=\\{X_{t}:t\in[S,T]\\}$ is a probability measure on
the path space

$W^{S,T}:=C([S,T];M),$

when $S=0$, we write $W^{T}:=W^{0,T}$ and $\Pi^{T}:=\Pi^{0,T}$ for simplicity.
For each point $x\in M$, let $X_{[0,T]}^{x}=\\{X_{t}^{x}:0\leq t\leq
T<T_{c}\\}$ and $W_{x}^{T}=\\{\gamma\in W^{T}:\gamma_{0}=x\\}$. It has been
well-known that there is a strong connection between the behavior of the
distribution of the $M$-valued Brownian motion associated with metric $g$ and
the geometry of their underlying space. This paper is devoted to further the
study of this relation over a inhomogeneous manifold. Note that although our
discussions base on the manifold with boundary, the results are also new for
the manifold without boundary.

When the metric is independent of $t$, the theory about stochastic analysis on
the path space over a complete Riemannian manifold has been well developed
since Driver [9] proved the quasi-invariance theorem for the Brownian motion.
Then, integration by parts formula for the associated gradient operator,
induced by the quasi-invariant flow, was established, which leads to the study
of the functional inequalities with respect to the corresponding Dirichlet
form (see e.g. [16, 12]). For the case with boundary, see [18, 25] for the
corresponding results on manifold with boundary; see [3, 5, 11] for an
intertwining formula for the differential of Itô development map.

A probabilistic approach to these problem was initiated by Abandon et al, who
constructed $g_{t}$-Brownian motion on time-inhomogeneous space in [2].
Recently, Chen [6] gives the interwining formula and log-Sobolev inequality
for usual Dirichlet form (without damp) on the path space over time-
inhomogeneous manifold. The main purpose of this paper is to prove the
integration by parts formula and further establish the log-Sobolev inequality
w.r.t. the associated Dirichlet form, defined by the damped gradient operator.
Note that from technical point of view, our method relies on [18, 25].

Our second purpose is to discuss the Talagrand type transportation-cost
inequalities on the path space with respect to the uniform distance on time-
inhomogeneous space. In 1996, Talagrand [20] found that the
$L^{2}$-Wasserstein distance to the standard Gaussian measure can be dominated
by the square root of twice relative entropy on $M=\mathbb{R}^{d}$ with
constant metric. This inequality has been extended to distributions on finite-
and infinite-dimensional spaces. In particular, this inequality was
established on the path space of diffusion processes with respect to several
different distances (i.e. cost functions). See [14] for the details on the
Wiener space with the Cameron-Martin distance; see [21, 8] on the path space
of diffusions with the $L^{2}$-distance; see [22] on the Riemannian path space
with intrinsic distance induced by the Malliavin gradient operator, and [13,
27, 24] on the path space of diffusions with the uniform distance.

The rest parts of the paper are organized as follows. In the following two
sections, we construct the Hsu’s multiplicative functional and then define the
corresponding damped gradient operator, which satisfies an integration by
parts formula induced by intrinsic quasi-invariant flows. In Section 4, the
log-Sobolev inequality for the associated Dirichlet form is established and
extends to the free path space. In Section 5, some transportation-cost
inequalities are presented to be equivalent to the curvature condition and the
convexity of the boundary, and parts of these are extended to non-convex case
in finial section.

##  2 (Reflecting) $L_{t}$-diffusion process and multiplicative functional

Let $\mathcal{F}(M)$ be the frame bundle over $M$ and $\mathcal{O}_{t}(M)$ be
the orthonormal frame bundle over $M$ with respect to $g_{t}$. Let
$\mathbf{p}:\mathcal{F}(M)\rightarrow M$ be the projection from
$\mathcal{F}(M)$ onto $M$. Let $\\{e_{\alpha}\\}_{\alpha=1}^{d}$ be the
canonical orthonormal basis of $\mathbb{R}^{d}$. For any $u\in\mathcal{F}(M)$,
let $H^{t}_{i}(u)$ be the $\nabla^{t}$ horizontal lift of $ue_{i}$ and
$\\{V_{\alpha,\beta}(u)\\}_{\alpha,\beta=1}^{d}$ be the canonical basis of
vertical fields over $\mathcal{F}(M)$, defined by
$V_{\alpha,\beta}(u)=Tl_{u}(\exp(E_{\alpha,\beta}))$, where $E_{\alpha,\beta}$
is the canonical basis of $\mathcal{M}_{d}(\mathbb{R})$, the $d\times d$
matric space over $\mathbb{R}$, and
$l_{u}:Gl_{d}(\mathbb{R})\rightarrow\mathcal{F}(M)$ is the left multiplication
from the general linear group to $\mathcal{F}(M)$, i.e.
$l_{u}\exp(E_{\alpha,\beta})=u\exp(E_{\alpha,\beta})$.

Let $B_{t}:=(B_{t}^{1},B_{t}^{2},\cdots,B_{t}^{d})$ be a
$\mathbb{R}^{d}$-valued Brownian motion on a complete filtered probability
space $(\Omega,\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$ with the natural
filtration $\\{\mathscr{F}_{t}\\}_{t\geq 0}$. Assume the elliptic generator
$L_{t}$ is a $C^{1}$ functional of time with associated metric $g_{t}$:

$L_{t}=\Delta_{t}+Z_{t}$

where $Z_{t}$ is a $C^{1,1}$ vector field on $M$. As in the time-homogeneous
case, to construct the $L_{t}$-diffusion process, we first construct the
corresponding horizontal diffusion process generated by
$L_{\mathcal{O}_{t}(M)}:=\Delta_{\mathcal{O}_{t}(M)}+H_{Z_{t}}^{t}$ by solving
the Stratonovich stochastic diffusion equation (SDE):

$\begin{cases}\mathrm{d}u_{t}=\sqrt{2}\displaystyle\sum_{i=1}^{d}H_{i}^{t}(u_{t})\circ\mathrm{d}B_{t}^{i}+H_{Z_{t}}^{t}(u_{t})\mathrm{d}t-\frac{1}{2}\displaystyle\sum_{i,j}\partial_{t}g_{t}(u_{t}e_{i},u_{t}e_{j})V_{i,j}(u_{t})\mathrm{d}t+H^{t}_{N_{t}}(u_{t})\mathrm{d}l_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ u_{0}\in\mathcal{O}_{0}(M),\end{cases}$

where $\Delta_{\mathcal{O}_{t}(M)}$ is the horizontal Laplace operator on
$\mathcal{O}_{t}(M)$; $H_{Z_{t}}^{t}$ and $H^{t}_{N_{t}}$ are the $\nabla^{t}$
horizontal lift of $Z_{t}$ and $N_{t}$ respectively; $l_{t}$ is an increasing
process supported on $\\{t\geq 0:X_{t}:={\bf p}u_{t}\in\partial M\\}$. By a
similar discussion as in [2, Proposition 1.2], we see that the last term
promises $u_{t}\in\mathcal{O}_{t}(M)$. Since $(H_{Z_{t}}^{t})_{t\in[0,T_{c})}$
is a $C^{1,1}$-family vector field, it is well-known that (see e.g. [17]) the
equation has a unique solution up to the life time
$\zeta:=\displaystyle\lim_{n\rightarrow\infty}\zeta_{n}$, and

$\zeta_{n}:=\inf\\{t\in[0,T_{c}):\rho_{t}({\bf p}u_{0},{\bf p}u_{t})\geq
n\\},\ n\geq 1,\ \ \inf\varnothing=T_{c},$

where $\rho_{t}(x,y)$ is the distance between $x$ and $y$ associated with
$g_{t}$. Let $X_{t}={\bf p}u_{t}$. It is easy to see that $X_{t}$ solves the
equation

$\mathrm{d}X_{t}=\sqrt{2}u_{t}\circ\mathrm{d}B_{t}+Z_{t}(X_{t})\mathrm{d}t+N_{t}(X_{t})\mathrm{d}l_{t},\
\ X_{0}=x:={\bf p}u$

up to the life time $\zeta$. By the Itô formula, for any $f\in
C_{0}^{1,2}([0,T_{c})\times M)$ with $N_{t}f_{t}:=N_{t}f_{t}|_{\partial M}=0$,

$f(t,X_{t})-f(0,x)-\int_{0}^{t}\left({\partial_{s}}+L_{s}\right)f(s,X_{s})\mathrm{d}s=\sqrt{2}\int_{0}^{t}\left<u_{s}^{-1}\nabla^{s}f(s,\cdot)(X_{s}),\mathrm{d}B_{s}\right>$

is a martingale up to the life time $\zeta$. Here and what follows, we denote
the inner product on $\mathbb{R}^{d}$ by $\left<\cdot,\cdot\right>$, and write
$f(t,\cdot)=f_{t}$ for simplicity. So, we call $X_{t}$ the reflecting
diffusion process generated by $L_{t}$. When $Z_{t}\equiv 0$, then
$\tilde{X}_{t}:=X_{t/2}$ is generated by $\frac{1}{2}\Delta_{t}$ and is called
the reflecting $g_{t}$-Brownian motion on $M$. In what follows, we assume the
process is non-explosive.

###  2.1 Multiplicative functional

To construct the desired continuous multiplicative functional, we need the
following assumption.

  1. (A)

There exist two constants $K,\sigma\in C([0,T_{c}))$ such that

$\mathcal{R}^{Z}_{t}\geq K(t),\ \mathbb{I}_{t}\geq\sigma(t)\ \mbox{and}\ \
\mathbb{E}e^{\lambda\int_{0}^{t}\sigma^{-}(s)\mathrm{d}l_{s}^{x}}<\infty$

holds for $\lambda>0$, $t\in[0,T_{c})$, and $x\in M$, where
$\sigma^{-}=0\vee(-\sigma)$.

To introduce Hsu’s discontinuous multiplicative functional, we need the lift
operators of $\mathcal{R}^{Z}_{t}$, ${\mathcal{G}}_{t}$, ${\mathbb{I}}_{t}$.
For any $u\in\mathcal{O}_{t}(M)$, let
$\mathcal{R}_{u}^{Z}(t)(a,b)=\mathcal{R}^{Z}_{t}(ua,ub)$ and
$\mathcal{G}_{u}(t)(a,b)=\mathcal{G}_{t}(ua,ub),\ a,b\in\mathbb{R}^{d}$. Let
${\bf p}_{\partial}:TM\rightarrow T\partial M$ be the orthogonal projection at
points on $(\partial M,g_{t})$. For any $u\in\mathcal{O}_{t}(M)$ with ${\bf
p}u\in\partial M$, let

$\mathbb{I}_{u}(t)(a,b)=\mathbb{I}_{t}({\bf p}_{\partial}ua,{\bf
p}_{\partial}ub),a,b\in\mathbb{R}^{d}.$

For $u\in\partial\mathcal{O}_{t}(M)$, the boundary of $\mathcal{O}_{t}(M)$,
let

${P}_{u}(t)(a,b)=\left<ua,N_{t}\right>_{t}\left<ub,N_{t}\right>_{t},a,b\in\mathbb{R}^{d}.$

For any $\varepsilon>0$ and $r\geq 0$, let $Q^{x,\varepsilon}_{r,t}$ solve the
following SDE on $\mathbb{R}^{d}\otimes\mathbb{R}^{d}$:

$\displaystyle\mathrm{d}Q^{x,\varepsilon}_{r,t}=-Q^{x,\varepsilon}_{r,t}\left\\{\mathcal{R}_{u^{x}_{t}}^{Z}(t)\mathrm{d}t+(\varepsilon^{-1}P_{u_{t}^{x}}(t)+\mathbb{I}_{u_{t}^{x}}(t))\mathrm{d}l^{x}_{t}\right\\}.$
(2.1)

When the metric is independent of $t$ and $M$ is compact, letting
$\varepsilon\downarrow 0$, the process $Q^{x,\varepsilon}_{r,t}$ converges in
$L^{2}$ to an adapted right-continuous process $Q^{x}_{r,t}$ with left limit,
such that $Q^{x}_{r,t}P_{u^{x}_{t}}(t)=0$ if $X^{x}_{t}\in\partial M$ (see
[18, Theorem 3.4]). Here, we follow Wang [26, Theorem 4.1.1], and introduce a
slightly different but simpler construction of the multiplicative functional
by solving a random integral equation on
$\mathbb{R}^{d}\otimes\mathbb{R}^{d}$.

######  Theorem 2.1.

Assume ${\bf(A)}$, then

  1. $(1)$

for any $x\in M$, $0\leq r\leq t<T_{c}$ and $u_{0}^{x}\in\mathcal{O}_{0}(M)$,
the equation

$Q^{x}_{r,t}=\left(I-\int_{r}^{t}Q_{r,s}^{x}\mathcal{R}^{Z}_{u_{s}^{x}}(s)\mathrm{d}s-\int_{s}^{t}Q_{r,s}^{x}\mathbb{I}_{u_{s}^{x}}(s)\mathrm{d}l^{x}_{s}\right)(I-{1}_{\\{X_{t}^{x}\in\partial
M\\}}P_{u_{t}^{x}}(t))$

has a unique solution;

  2. $(2)$

for any $0\leq r\leq t<T_{c}$, $\|Q_{r,t}^{x}\|\leq
e^{-\int_{r}^{t}K(s)\mathrm{d}s-\int_{r}^{t}\sigma(s)\mathrm{d}l^{x}_{s}}$
a.s., where $\|\cdot\|$ is the operator norm for $d\times d$-matrices;

  3. $(3)$

for any $0\leq r\leq s\leq t<T_{c}$, $Q^{x}_{r,t}=Q^{x}_{r,s}Q^{x}_{s,t}$
a.s..

###### Proof.

We only consider the existence of the solution up to a arbitrarily given time
$T\in(r,T_{c})$, since the uniqueness is obvious. In the following, we drop
the superscript $x$ for simplicity. By the assumption $\mathbf{(A)}$ and
(2.1), we have

$\displaystyle\|Q_{r,t}^{\varepsilon}\|^{2}\leq
1-2\int_{r}^{t}\|Q_{r,s}^{\varepsilon}\|^{2}K(s)\mathrm{d}s-2\int_{r}^{t}\|Q_{r,s}^{\varepsilon}\|^{2}\sigma(s)\mathrm{d}l_{s}-\frac{2}{\varepsilon}\int_{s}^{t}\|Q_{r,s}^{\varepsilon}P_{u_{s}}(s)\|^{2}\mathrm{d}l_{s},\
\ t>r.$

Therefore, we obtain $\|Q_{r,t}^{\varepsilon}\|^{2}\leq
e^{-2\int_{r}^{t}K(s)\mathrm{d}s-2\int_{r}^{t}\sigma(s)\mathrm{d}l_{s}},\ \
t\geq r$ and

$\displaystyle\int_{r}^{T}\|Q^{\varepsilon}_{r,s}P_{u_{s}}(s)\|^{2}\mathrm{d}l_{s}\leq\frac{\varepsilon}{2}\left[1+2\left(\int_{r}^{T}K^{-}(s)\mathrm{d}s+\int_{r}^{T}\sigma^{-}(s)\mathrm{d}l_{s}\right)e^{2\int_{r}^{T}K^{-}(s)\mathrm{d}s+\int_{r}^{T}\sigma^{-}(s)\mathrm{d}l_{s}}\right].$
(2.2)

Combining this with ${\bf(A)}$, we obtain

$\displaystyle\lim_{\varepsilon\rightarrow
0}\mathbb{E}\int_{r}^{T}\|Q_{r,s}^{\varepsilon}P_{u_{s}}(s)\|^{2}\mathrm{d}l_{s}=0$
(2.3)

and

$\sup_{\varepsilon\in(0,1)}\mathbb{E}\int_{r}^{T}\|Q_{r,t}^{\varepsilon}\|^{2}(\mathrm{d}t+\mathrm{d}l_{t})<\infty.$

Because of the latter, we may select a sequence $\varepsilon_{n}\downarrow 0$
and an adapted process $\overline{Q}_{r,\cdot}\in
L^{2}(\Omega\times[r,T])\rightarrow\mathbb{R}^{d}\otimes\mathbb{R}^{d};\mathbb{P}\times(\mathrm{d}t+\mathrm{d}l_{t})$,
such that for any $g\in
L^{2}(\Omega\times[r,T])\rightarrow\mathbb{R}^{d};\mathbb{P}\times(\mathrm{d}t+\mathrm{d}l_{t})$,

$\lim_{n\rightarrow\infty}\mathbb{E}\int_{r}^{T}Q^{\varepsilon_{n}}_{r,t}g_{t}(\mathrm{d}t+\mathrm{d}l_{t})=\mathbb{E}\int_{r}^{T}\overline{Q}_{r,t}g_{t}(\mathrm{d}t+\mathrm{d}l_{t}).$

The following discussion is almost the same as the case with constant metric,
see [25] for details. ∎

The following result is a direct conclusion of Theorem 2.1.

######  Proposition 2.2.

Assume (A). For any $\mathbb{R}^{d}$-valued continuous semi-martingale $h_{t}$
with

$1_{\\{X^{x}_{t}\in\partial M\\}}P_{u_{t}}(t)h_{t}=0,$
$\mathrm{d}Q^{x}_{r,t}h_{t}=\overline{Q}^{x}_{r,t}\mathrm{d}h_{t}-Q^{x}_{r,t}\mathcal{R}_{u_{t}^{x}}^{Z}(t)h_{t}\mathrm{d}t-Q^{x}_{r,t}\mathbb{I}_{u_{t}^{x}}(t)h_{t}\mathrm{d}l^{x}_{t},\
\ t\geq r,$

where

$\overline{Q}^{x}_{r,t}=\left(I-\int_{r}^{t}Q^{x}_{r,s}\mathcal{R}^{Z}_{u_{s}^{x}}(s)\mathrm{d}s-\int_{r}^{t}Q^{x}_{r,s}\mathbb{I}_{u_{s}^{x}}(s)\mathrm{d}l_{s}^{x}\right).$

###### Proof.

As $1_{\\{X^{x}_{t}\in\partial M\\}}P_{u_{t}}(t)h_{t}=0,$ and by Theorem 2.1,
we have

$Q^{x}_{r,t}h_{t}=\left(I-\int_{r}^{t}Q_{r,s}^{x}\mathcal{R}_{u_{s}^{x}}^{Z}(s)\mathrm{d}s+\frac{1}{2}\int_{r}^{t}Q_{r,s}^{x}\mathcal{G}_{u_{s}^{x}}(s)\mathrm{d}s-\int_{r}^{t}Q_{r,s}\mathbb{I}_{u_{s}^{x}}(s)\mathrm{d}l_{s}^{x}\right)h_{t}=\overline{Q}_{r,t}^{x}h_{t}.$

Then the proof is completed by using Itô formula. ∎

Recall that $\\{P_{r,t}\\}_{0\leq r\leq t<T_{c}}$ is the Neumann semigroup
generated by $L_{t}$. The following is a consequence of Proposition 2.2, which
is the derivative formula of the diffusion semigroup, known as Bismut-Elworthy
Li formula (see e.g. [3, 10]).

######  Corollary 2.3.

Assume ${\bf(A)}$. Let $f\in C^{\infty}_{b}(M)$. Then for any $0\leq
r<t<T_{c}$,

$[r,t]\ni s\rightarrow
Q^{x}_{r,s}(u^{x}_{s})^{-1}\nabla^{s}P_{s,t}f(X^{x}_{s})$

is a martingale. Consequently,

$\displaystyle(u_{r}^{x})^{-1}\nabla^{r}P_{r,t}f(X_{r}^{x})=\mathbb{E}(Q^{x}_{r,t}(u_{t}^{x})^{-1}\nabla^{t}f(X_{t}^{x})|\mathscr{F}_{r}),$
(2.4)

and for any adapted $\mathbb{R}_{+}$-valued precess $\xi$ satisfying
$\xi(r)=0,\xi(t)=1$, and
$\mathbb{E}(\int_{r}^{t}\xi^{\prime}(s)^{2}\mathrm{d}s)^{\alpha}<\infty$ for
$\alpha>1/2$, there holds

$(u_{r}^{x})^{-1}\nabla^{r}P_{r,t}f(X_{r}^{x})=\frac{1}{\sqrt{2}}\mathbb{E}\left(f(X_{t}^{x})\int_{r}^{t}\xi^{\prime}(s)(Q_{r,s}^{x})^{*}\mathrm{d}B_{s}\big{|}\mathscr{F}_{r}\right).$

###### Proof.

The proof is essentially due to [25, Corollary 3.4]. Without losing
generality, we assume $r=0$ and drop the superscript $x$ for simplicity.

We first prove that $Q_{s}h_{s}$ is a martingale. Let
$h_{s}=(u_{s})^{-1}\nabla^{s}f(X_{s})$. Since $\nabla^{s}P_{s,t}f$ is vertical
to $N_{s}$ on $\partial M$, $1_{\\{X_{s}\in\partial M\\}}P_{u}(s)h_{s}=0$.
Then we have

$\mathrm{d}Q_{s}h_{s}=\overline{Q}_{s}\mathrm{d}h_{s}-Q_{s}\mathcal{R}_{u}^{Z}(s)h_{s}\mathrm{d}s-Q_{s}\mathbb{I}_{u}(s)h_{s}\mathrm{d}l_{s}.$

Let $F(u,s):=u^{-1}\nabla^{s}P_{s,t}f(\mathbf{p}u),\ u\in\mathcal{O}_{s}(M)$,
then

$\frac{\mathrm{d}}{\mathrm{d}s}F(u,s)=-u^{-1}\nabla^{s}P_{s,t}f({\bf
p}u)=-L_{\mathcal{O}_{s}(M)}F(\cdot,s)(u)+(\mathcal{R}_{u}^{Z}(s)+\frac{1}{2}\mathcal{G}_{u}(s))F(u,s),s\in[0,t].$

On the other hand, noting that

$\mathrm{d}u_{t}=\sqrt{2}\sum_{i=1}^{d}H_{i}^{t}\circ\mathrm{d}B_{t}^{i}+H^{t}_{Z_{t}}(u_{t})\mathrm{d}t-\frac{1}{2}\sum_{\alpha,\beta=1}^{d}\mathcal{G}_{\alpha,\beta}(t,u_{t})V_{\alpha,\beta}(u_{t})\mathrm{d}t+H^{t}_{N_{t}}(u_{t})\mathrm{d}l_{t}.$

By the Itô formula, we have

$\displaystyle\mathrm{d}F(u_{s},t_{0})=$
$\displaystyle\mathrm{d}M_{s}+L_{\mathcal{O}_{s}{M}}F(\cdot,t_{0})(u_{s})\mathrm{d}s+H_{N_{s}}^{s}F(\cdot,t_{0})(u_{s})\mathrm{d}l_{s}$
$\displaystyle-\frac{1}{2}\sum_{\alpha,\beta}\mathcal{G}_{\alpha,\beta}(s,u_{s})V_{\alpha,\beta}(u_{s})F(\cdot,t_{0})(u_{s})\mathrm{d}s.$
(2.5)

where

$\mathrm{d}M_{s}:=\sqrt{2}\sum_{i=1}^{d}H^{s}_{i}F(\cdot,t_{0})(u_{s})\mathrm{d}B_{s}^{i}.$

Therefore,

$\mathrm{d}h_{s}=\mathrm{d}M_{s}+\mathcal{R}_{u}^{Z}(s)h_{s}\mathrm{d}s+H_{N_{s}}^{s}F(\cdot,s)(u_{s})\mathrm{d}l_{s}.$

Since $1_{\\{X_{s}\in\partial M\\}}Q_{s}P_{u_{s}}(s)=0$, combining this with
(2.1), we obtain

$\mathrm{d}Q_{s}h_{s}=Q_{s}\mathrm{d}M_{s}+Q_{s}(I-P_{u_{s}}(s))\left\\{H_{N_{s}}^{s}F(\cdot,s)(u_{s})-\mathbb{I}_{u}{(s)}F(u_{s},s)\right\\}\mathrm{d}l_{s}.$

Noting that for any $e\in\mathbb{R}^{d}$, it follows from that when
$X_{s}\in\partial M$,

$\displaystyle\left<(I-P_{u_{s}}(s))H^{s}_{N_{s}}F(\cdot,s)(u_{s}),e\right>$
$\displaystyle={\rm
Hess}^{s}_{P_{s,t}f}(N_{s},\mathbf{p}_{\partial}u_{s}e)=\mathbb{I}_{s}(\nabla^{s}P_{s,t}f(X_{s}),\mathbf{p}_{\partial}u_{s}e)$
$\displaystyle=\mathbb{I}_{u}(s)(F(u,s),e)=\left<\mathbb{I}_{u}(s)F(u,s),e\right>,$

we conclude that

$(1-P_{u}(s))\\{H^{s}_{N_{s}}F(\cdot,s)(u_{s})-\mathbb{I}_{u}(s)F(u_{s},s)\\}\mathrm{d}l_{s}=0.$

Therefore, $Q_{s}h_{s}$ is a local martingale. By (A), it is then a martingale
according to [7, Theorem 4.9].

The following step is similar as shown in step (b) in the proof of [25, Lemma
3.3]. We skip it here.

∎

##  3 Damped Gradient, quasi-invariant flows and integration by Parts

When the metric is independent of $t$, the Malliavin derivative can be
realized by quasi-invariant flows for diffusion on manifolds (see e.g. [17,
25]). In this section, by using the multiplicative functional constructed in
§2.1, we first introduce the damped gradient operator as in [11], then
introduce quasi-invariant flows induced by SDEs with refection, and finally
link them by establishing an integration by parts formula.

###  3.1 Damped Gradient operator and quasi-invariant flows

We shall use multiplicative functionals $\\{Q^{x}_{r,t}:0\leq r\leq
t<T_{c}\\}$ to define the damped gradient operator for functionals of $X^{x}$.

Let

$\mathscr{F}C_{0}^{\infty}=\\{W^{T}\ni\gamma\rightarrow
f(\gamma_{t_{1}},\gamma_{t_{2}},\cdots,\gamma_{t_{n}}):n\geq
1,0<t_{1}<t_{2}<\cdots<t_{n}\leq T,f\in C_{0}^{\infty}(M^{n})\\}$

be the class of smooth cylindrical functions on $W^{T}$. Let

${\bf H}_{0}:=\left\\{h\in C([0,T];\mathbb{R}^{d}):h(0)=0,\|h\|_{{\bf
H}_{0}}:=\int_{0}^{T}|\dot{h}(s)|^{2}\mathrm{d}s<\infty\right\\}$

be the Cameron-Martin space on the flat path space. For any
$F\in\mathscr{F}C^{\infty}_{0}$ with
$F(\gamma)=f(\gamma_{t_{1}},\gamma_{t_{2}},\cdots,\gamma_{t_{n}})$, define the
damped gradient $D^{0}F(X_{[0,T]}^{x})$ as an ${\bf H}_{0}$-valued random
variable by setting $(D^{0}F(X^{x}_{[0,T]}))(0)=0$ and

$\frac{\mathrm{d}}{\mathrm{d}t}(D^{0}F(X^{x}_{[0,T]}))(t)=\sum_{i=1}^{n}1_{\\{t<t_{i}\\}}Q^{x}_{t,t_{i}}(u^{x}_{t_{i}})^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x}),\
t\in[0,T].$

where $\nabla^{t_{i}}_{i}$ denotes the gradient operator w.r.t. the $i$-th
component associated with $g_{t_{i}}$. Then, for any ${\bf H}_{0}$-valued
random variable $h$, let

$D^{0}_{h}F(X_{[0,T]}^{x})=\left<D^{0}F(X_{[0,T]}),h\right>_{{\bf
H_{0}}}=\sum_{i=1}^{n}\int_{0}^{t_{i}}\left<(u_{t_{i}}^{x})^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x}),(Q_{t,t_{i}}^{x})^{*}h^{\prime}(t)\right>\mathrm{d}t,$

We would like to indicate that the formulation of $D^{0}_{h}F$ is consistent
with [11] for the case with constant metric. Note that compared with usual
gradient operator, it contains the multiplicative functional, which affects
the log-Sobolev constant. This operator links $D^{0}_{h}F$ to the directional
derivative induced by a quasi-invariant flow. We now turn to investigating
this relation. The main idea essentially due to [18], where quasi-invariant
flows are constructed for constant manifold $M$ with boundary $\partial M$.
Let ${\bf\tilde{H}_{0}}$ be the set of all adapted elements in
$L^{2}(\Omega\rightarrow\mathbf{H}_{0};\mathbb{P})$; i.e.

${\bf\tilde{H}_{0}}=\\{h\in
L^{2}(\Omega\rightarrow{\bf{H}_{0}};\mathbb{P}):h(t)\ \mbox{is\ a}\
\mathscr{F}_{t}\mbox{-measurable},t\in[0,T]\\}.$

Then, ${\bf\tilde{H}_{0}}$ is a Hilbert space with inner product

$\big{<}h,\tilde{h}\big{>}_{{\bf\tilde{H}_{0}}}:=\mathbb{E}\big{<}h,\tilde{h}\big{>}_{{\bf
H_{0}}}=\mathbb{E}\int_{0}^{T}\big{<}h^{\prime}(t),\tilde{h}^{\prime}(t)\big{>}\mathrm{d}t,h,\
\tilde{h}\in{{\bf\tilde{H}_{0}}}.$

For $h\in\mathbf{\tilde{H}_{0}}$ and $\varepsilon>0$, let
$X^{\varepsilon,h}_{t}$ solve the SDE

$\displaystyle\mathrm{d}X_{t}^{\varepsilon,h}=\sqrt{2}u_{t}^{\varepsilon,h}\circ\mathrm{d}B_{t}+Z_{t}(X_{t}^{\varepsilon,h})\mathrm{d}t++\varepsilon\sqrt{2}u_{t}^{\varepsilon,h}h^{\prime}(t)\mathrm{d}t+N_{t}(X_{t}^{\varepsilon,h})\mathrm{d}l_{t}^{\varepsilon,h},$
(3.1)

where $l_{t}^{\varepsilon,h}$ and $u_{t}^{\varepsilon,h}$ are, respectively,
the local time on $\partial M$ and the horizontal lift on $\mathcal{O}_{t}(M)$
for $X_{t}^{\varepsilon,h}$. The detailed construction of
$X^{\varepsilon,h}_{t}$ is similar as in Section 2. To see that
$\\{X_{[0,T]}^{\varepsilon,h}\\}_{\varepsilon\geq 0}$ has the flow property
also in our setting, let

$\Theta:W_{0}:=\\{\omega\in C([0,T];\mathbb{R}^{d}):\omega_{0}=0\\}\rightarrow
W^{T}$

be measurable such that $X=\Theta(B)$, $B\in W_{0}$. For any $\varepsilon>0$
and a function $\Phi:W_{0}\rightarrow W^{T}$, let
$(\theta^{h}_{\varepsilon}\Phi)(\omega)=\Phi(\omega+\varepsilon h)$. Then
$X_{[0,T]}^{\varepsilon,h}=(\theta^{h}_{\varepsilon}\Theta)(B),\
\varepsilon>0$. Hence,

$X^{\varepsilon_{1}+\varepsilon_{2},h}_{[0,T]}=\theta^{h}_{\varepsilon_{1}}X^{\varepsilon_{2},h}_{[0,T]},\
\varepsilon_{1},\varepsilon_{2}\geq 0.$

Moreover, let us explain that the flow is quasi-invariant, i.e. for each
$\varepsilon\geq 0$, the distribution of $X^{\varepsilon,h}_{[0,T]}$ is
absolutely continuous w.r.t. that of $X^{x}_{[0,T]}$. Let

$R^{\varepsilon,h}=\exp{\left[\varepsilon\int_{0}^{T}\left<h^{\prime}(t),\mathrm{d}B_{t}\right>-\frac{\varepsilon^{2}}{2}\int_{0}^{T}|h^{\prime}(t)|^{2}\mathrm{d}t\right]}.$

By the Girsanov theorem

$B_{t}^{\varepsilon,h}:=B_{t}-\varepsilon h(t)$

is the $d$-dimensional Brownian motion under the probability
$R^{\varepsilon,h}\mathbb{P}$. Thus, the distribution of $X^{x}_{[0,T]}$ under
$R^{\varepsilon,h}$ coincides with that of $X^{\varepsilon,h}_{[0,T]}$ under
$\mathbb{P}$. Therefore, the map $X^{x}_{[0,T]}\rightarrow
X^{\varepsilon,h}_{[0,T]}$ is quasi-invariant. The quasi-invarient property
leads us to prove the following property.

######  Proposition 3.1.

Let $x\in M$ and $F\in\mathscr{F}C^{\infty}$. Then

$\displaystyle\lim_{\varepsilon\downarrow
0}\mathbb{E}\frac{F(X^{\varepsilon,h}_{[0,T]})-F(X_{[0,T]}^{x})}{\varepsilon}=\mathbb{E}\left\\{F(X_{[0,T]}^{x})\int_{0}^{T}\left<h^{\prime}(t),\mathrm{d}B_{t}\right>\right\\}$

holds for $h\in{\bf\tilde{H}}_{0,b}$, the set of all elements in
${\bf\tilde{H}}_{0}$ with bounded $\|h\|_{\tilde{\mathbf{H}}_{0}}$.

###### Proof.

As we have explained that $B_{t}^{\varepsilon,h}=B_{t}-\varepsilon h(t)$ is a
$d$-dimensional Brownian motion under $R^{\varepsilon,h}\mathbb{P}$. By the
weak uniqueness of (3.1), we conclude that the distribution of $X^{x}$ under
$R^{\varepsilon,h}\mathbb{P}$ coincide with that of $X^{\varepsilon,h}$ under
$\mathbb{P}$. In particular,
$\mathbb{E}F(X_{[0,T]}^{\varepsilon,h})=\mathbb{E}[R^{\varepsilon,h}F(X_{[0,T]}^{x})]$.
Thus, the assertion follows from
$\frac{\mathrm{d}R^{\varepsilon,h}}{\mathrm{d}\varepsilon}|_{\varepsilon=0}$
and the dominated convergence theorem since
$\\{R^{\varepsilon,h}\\}_{\varepsilon\in[0,1]}$ is uniformly integrable for
$h\in\tilde{{\bf H}}_{0,b}$. ∎

###  3.2  Integration by parts formula

In this section, an integration by parts formula for $D^{0}_{h}F$ is
established and applied to clarifying the link between this formula and the
derivative induced by the flow
$\\{X^{\varepsilon,h}_{[0,T]}\\}_{\varepsilon\geq 0}.$ The main result of this
subsection is presented as follows.

######  Theorem 3.2.

Assume ${\bf(A)}$. For any $x\in M$ and $F\in\mathscr{F}C_{0}^{\infty}$,

$\displaystyle\lim_{\varepsilon\downarrow
0}\mathbb{E}\frac{F(X^{\varepsilon,h}_{[0,T]})-F(X^{x}_{[0,T]})}{\varepsilon}=\mathbb{E}\\{D^{0}_{h}F\\}(X^{x}_{[0,T]})=\mathbb{E}\left\\{F(X^{x}_{[0,T]})\int_{0}^{T}\left<h^{\prime}(t),\mathrm{d}B_{t}\right>\right\\}$
(3.2)

holds for all $h\in{\bf\tilde{H}}_{0,b}$.

###### Proof.

By the Proposition 3.1, it is sufficient for us to prove the second equality.
The proof is similar to the constant metric case (see [25, Theorem 2.1]) due
to the following Lemmas 3.3 and 3.5 and the Markov property, . Note that the
second equality holds for all $h\in\tilde{{\bf H}}_{0}$. ∎

The following lemma gives the gradient formula for special cylinder functions.

######  Lemma 3.3.

For any $n\geq 1$, $0<t_{1}<\cdots<t_{n}\leq T$, and $f\in C^{\infty}(M^{n})$,

$\displaystyle(u_{t_{1}}^{x})^{-1}\nabla^{t_{1}}_{1}\mathbb{E}\\{f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x})|\mathscr{F}_{t_{1}}\\}=\sum_{i=1}^{n}\mathbb{E}\\{Q_{t_{1},t_{i}}^{x}(u_{t_{i}}^{x})^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x})|\mathscr{F}_{t_{1}}\\}$

holds for all $x\in M$ and $u_{0}^{x}\in\mathcal{O}_{0}(M)$, where
$\nabla^{t_{i}}_{i}$ denotes the $g_{t_{i}}$-gradient w.r.t. the $i$-th
component.

###### Proof.

It is obvious that the assertion is true for $n=1$. By (2.4), we have

$(u_{t_{1}}^{x})^{-1}\nabla^{t_{1}}\mathbb{E}(f(X_{t_{2}})|\mathscr{F}_{t_{1}})=\mathbb{E}(Q_{t_{1},t_{2}}^{x}(u_{t_{2}}^{x})^{-1}\nabla_{2}^{t_{2}}f(X_{t_{2}})|\mathscr{F}_{t_{1}}).$

which plus the case of $n=1$, we prove the result for $n=2$. Assume that it
holds for $n=k$, $k\geq 2$. It remains to prove the case for $n=k+1$. To this
end, by Markov property, set

$g(X_{t_{1}}^{x},X_{t_{2}}^{x})=\mathbb{E}f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{k+1}}^{x}|\mathscr{F}_{t_{2}})$

By the assumption for $n=k$, we have

$\displaystyle(u_{t_{1}}^{x})^{-1}\nabla^{t_{1}}_{1}\mathbb{E}\left\\{g(X_{t_{1}},X_{t_{2}})|\mathscr{F}_{t_{1}}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{(u_{t_{1}}^{x})^{-1}\nabla_{1}^{t_{1}}g(X_{t_{1}}^{x},X_{t_{2}}^{x})|\mathscr{F}_{t_{1}}\right\\}+\mathbb{E}\\{Q_{t_{1},t_{2}}(u_{t_{2}}^{x})^{-1}\nabla^{t_{2}}_{2}g(X_{t_{1}},X_{t_{2}})|\mathscr{F}_{t_{1}}\\}$
(3.3)

for $x\in M$ and $u_{0}\in\mathcal{O}_{0}(M)$. Fix the value of
$X_{t_{1}}^{x}=x_{0}$, by the assumption for $n=k$, we have

$(u_{t_{2}}^{x})^{-1}\nabla^{t_{2}}_{2}\mathbb{E}(f(x_{0},X_{t_{2}}^{x},\cdots,X_{t_{k+1}}^{x})|\mathscr{F}_{t_{2}})=\sum_{i=2}^{k+1}\mathbb{E}\left\\{Q_{t_{2},t_{i}}^{x}(u_{t_{i}}^{x})^{-1}\nabla_{i}^{t_{i}}f(x_{0},X^{x}_{t_{2}},\cdots,X^{x}_{t_{k+1}})|\mathscr{F}_{t_{2}}\right\\}.$

Combining this with (3.2), we have

$\displaystyle(u_{t_{1}}^{x})^{-1}\nabla^{t_{1}}_{1}\mathbb{E}\left\\{f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X^{x}_{t_{k+1}})\bigg{|}\mathscr{F}_{t_{1}}\right\\}=(u_{t_{1}}^{x})^{-1}\nabla^{t_{1}}_{1}\mathbb{E}\\{g(X^{x}_{t_{1}},X^{x}_{t_{2}})|\mathscr{F}_{t_{1}}\\}$
$\displaystyle=\sum_{i=1}^{k+1}\mathbb{E}\left\\{Q^{x}_{t,t_{i}}(u^{x}_{t_{i}})^{-1}\nabla^{t_{i}}_{i}f(X^{x}_{t_{1}},X^{x}_{t_{2}},\cdots,X^{x}_{t_{k+1}})|\mathscr{F}_{t_{1}}\right\\}.$
(3.4)

∎

######  Remark 3.4.

Especially, choosing $t_{1}=0$, we arrive at

$\displaystyle(u_{0}^{x})^{-1}\nabla^{0}\mathbb{E}\\{f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x})\\}=\sum_{i=1}^{n}\mathbb{E}\left\\{Q^{x}_{t_{i}}(u_{t_{i}}^{x})^{-1}\nabla_{i}^{t_{i}}f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x})\right\\}.$
(3.5)

The following result is a direct consequence of (2.4) and the Itô formula for
$f(X_{t}^{x})$, i.e.

$f(X_{t}^{x})=f(x)+\sqrt{2}\int_{0}^{t}\left<(u_{s}^{x})^{-1}\nabla^{s}P_{s,t}f(X_{s}^{x}),\mathrm{d}B_{s}\right>.$

######  Lemma 3.5.

For any $n\geq 1$, $0<t_{1}<t_{2}<\cdots<t_{n}\leq T$, and $f\in
C^{\infty}(M^{n})$,

$\displaystyle\mathbb{E}\left\\{f(X_{t}^{x})\int_{0}^{t}\left<h_{s}^{\prime},\mathrm{d}B_{s}\right>\right\\}=\mathbb{E}\int_{0}^{t}\left<(u_{t}^{x})^{-1}\nabla^{t}f(X_{t}^{x}),(Q^{x}_{s,t})^{*}h_{s}\right>\mathrm{d}s,h\in\mathbf{\tilde{H}}_{0},t\in[0,T]$

holds for all $x\in M$ and $u_{0}^{x}\in\mathcal{O}_{0}(M)$.

##  4  The Log-Sobolev Inequality

When the metric is independent of $t$, log-Sobolov inequalities on $W_{x}^{T}$
were established independently by Hsu [18] and by Aida and Elworthy [1]. In
this section, we first consider the path space with a fixed initiated point,
then move to the free path space following an idea of [11], where the (non-
damped) gradient operator is studied on the free path space over the constant
manifolds without boundary.

###  4.1  Log-Sobolev inequality on $W^{T}_{x}$

Let $\Pi^{T}_{x}$ be the distribution of $X^{x}_{[0,T]}$. Let

$\mathscr{E}^{x}(F,G)=\mathbb{E}\left\\{\left<D^{0}F,D^{0}G\right>_{\bf
H_{0}}(X^{x}_{[0,T]})\right\\},\ F,\ G\in\mathscr{F}C_{0}^{\infty}.$

Since both $D^{0}F$ and $D^{0}G$ are functionals of $X$,
$(\mathscr{E}^{x},\mathscr{F}C_{0}^{\infty})$ is a positive bilinear form on
$L^{2}(W^{T}_{x};\Pi^{T}_{x})$. It is standard that the integration by parts
formula (3.2) implies the closability of the form, see Lemma 4.1. We shall use
$(\mathscr{E}^{x},\mathscr{D}(\mathscr{E}^{x}))$ to denote the closure of
$(\mathscr{E}^{x},\mathscr{F}C^{\infty}_{0})$. Moreover, (3.2) also implies
the Clark-Ocône type martingale representation formula, see Lemma 4.2, which
leads to the standard Gross log-Sobolev inequality (see e.g. [15]).

######  Lemma 4.1.

Assume ${\bf(A)}$. $(\mathscr{E}^{x},\mathscr{F}C_{0}^{\infty})$ is closable
in $L^{2}(W^{T}_{x};\Pi^{T}_{x})$.

###### Proof.

By the integration by part formula, the discussion is standard (see [25, Lemma
4.1]), we omit it here. ∎

The following result gives us the Clark-Ocône type martingale representation
formula for $F(X_{[0,T]})$. Following the proof of [25, Lemma 4.2] for the
case with constant metric, we have

######  Lemma 4.2.

(Clark-Haussman-Ocône Formula) Assume ${\bf(A)}$. For any
$F\in\mathscr{F}C_{0}^{\infty}$, let $\tilde{D}^{0}F(X_{[0,T]}^{x})$ be the
projection of $D^{0}F(X_{[0,T]}^{x})$ on $\mathbf{\tilde{H}}_{0}$, i.e.

$\frac{\mathrm{d}}{\mathrm{d}t}(\tilde{D}^{0}F(X_{[0,T]}^{x}))(t)=\mathbb{E}\left(\frac{\mathrm{d}}{\mathrm{d}t}(D^{0}F(X_{[0,T]}^{x}))\big{|}\mathscr{F}_{t}\right),\
t\in[0,T],\ (\tilde{D}^{0}F(X_{[0,T]}^{x}))(0)=0.$

Then

$F(X_{[0,T]}^{x})=\mathbb{E}F(X^{x}_{[0,T]})+\int_{0}^{T}\left<\frac{\mathrm{d}}{\mathrm{d}t}(\tilde{D}^{0}F(X_{[0,T]}^{x})(t),\mathrm{d}B_{t}\right>.$

It is standard that the martingale representation in Lemma 4.2 implies the
following log-Sobolev inequality. Since the parameter $T$ and the information
of Ricci curvature and the second fundamental form have been properly
contained in the Dirichlet form $\mathscr{E}$, the resulting log-Sobolev
constant is independent of $T$, $K$ and $\sigma$. Moreover, it is well-known
that the constant 2 in the inequality is sharp constant for compact manifolds
with constant metric.

######  Theorem 4.3.

Assume $\bf{(A)}$. For any $T>0$ and $x\in M$,
$(\mathscr{E}^{x},\mathscr{D}(\mathscr{E}^{x}))$ satisfies the following log-
Sobolev inequality,

$\Pi^{T}_{x}(F^{2}\log F^{2})\leq 2\mathscr{E}^{x}(F,F),\
F\in\mathscr{D}(\mathscr{E}^{x}),\ \Pi_{x}^{T}(F^{2})=1.$

###### Proof.

Due to lemma 4.1, it sufficient to prove the inequality for
$F\in\mathscr{F}C_{0}^{\infty}$. Let

$m_{t}:=\mathbb{E}\left(F(X^{x}_{[0,T]})^{2}\big{|}\mathscr{F}_{t}\right),\ \
t\in[0,T].$

By the Itô formula,

$\displaystyle\mathrm{d}m_{t}\log m_{t}=(1+\log
m_{t})\mathrm{d}m_{t}+\frac{|\frac{\mathrm{d}}{\mathrm{d}t}(\tilde{D}^{0}F(X_{[0,T]}^{x}))(t)|^{2}}{2m_{t}}\mathrm{d}t.$

Therefore,

$\displaystyle\Pi_{x}^{T}(F^{2}\log F^{2})$
$\displaystyle=\mathbb{E}_{x}m_{T}\log
m_{T}=\int_{0}^{T}\frac{2\mathbb{E}\left(F(X_{[0,T]}^{x})\frac{\mathrm{d}}{\mathrm{d}t}(D^{0}F(X_{[0,T]}^{x})(t))\
\big{|}\mathscr{F}_{t}\right)^{2}}{\mathbb{E}\left(F(X_{[0,T]}^{x})^{2}|\mathscr{F}_{t}\right)}\mathrm{d}t$
$\displaystyle\leq
2\int_{0}^{T}\mathbb{E}\left|\frac{\mathrm{d}}{\mathrm{d}t}(D^{0}F(X_{[0,T]}^{x}))(t)\right|^{2}\mathrm{d}t$
$\displaystyle=2\mathbb{E}\|D^{0}F(X_{[0,T]}^{x})\|^{2}_{{\bf H}_{0}}$
$\displaystyle=2\mathscr{E}^{x}(F,F).$

∎

Note that on manifolds without boundary equipped with time-depending metric,
the log-Sobolov inequality with respect to the Dirichlet form induced by usual
gradient derivative is recently established in [6]. And the log-Sobolov
constant is $2\exp\\{\sup_{t\in[0,T]}|\mathcal{R}_{t}^{Z}|\\}$.

###  4.2 Application to free path spaces

Let $\Pi^{T}_{\mu}$ be the distribution of the (reflecting) diffusion process
generated by $L_{t}:=\nabla^{t}+Z_{t}$ with initial distribution $\mu$ and
time-interval $[0,T]$. Due to the freedom of the initial point, it is natural
for us to make use of the following Cameron-Martin space:

${\bf H}=\left\\{h\in
C([0,T];\mathbb{R}^{d}):\int_{0}^{T}|h^{\prime}(t)|^{2}\mathrm{d}t<\infty\right\\}.$

Then ${\bf H}$ is a Hilbert space under the inner product

$\left<h_{1},h_{2}\right>_{\bf
H}=\left<h_{1}(0),h_{2}(0)\right>+\int_{0}^{T}\left<h^{\prime}_{1}(t),h^{\prime}_{2}(t)\right>\mathrm{d}t.$

To defined the damped gradient operator on the free path space, let

$\overline{\Omega}=M\times\Omega,\
\overline{\mathscr{F}}_{t}=\mathscr{B}(M)\times\mathscr{F}_{t},\ \mbox{and}\
\overline{\mathbb{P}}=\mu\times\mathbb{P}.$

Let $X_{0}(x,\omega)=x$ for $(x,\omega)\in M\times\Omega$. Then, under the
filtered probability space $(\overline{\Omega},\ \overline{\mathscr{F}}_{t},\
\overline{\mathbb{P}})$, $X_{t}(x,\omega):=X^{x}_{t}(\omega)$ is the
(reflecting) diffusion process generated by $L_{t}$ stating from $x$, and
$u_{t}(x,\omega):=u^{x}_{t}(\omega)$ is its horizontal lift. Moreover, let
$Q_{r,t}(x,\omega)=Q^{x}_{r,t}(\omega)$ for $0\leq r\leq t$, and write
$Q_{t}^{x}(\omega):=Q_{0,t}(x,\omega)$ for simplicity. Now, for any $F\in
FC^{\infty}_{0}$ with $F(\gamma)=f(\gamma_{t_{1}},\cdots,\gamma_{t_{n}})$, let

$\displaystyle
DF(X)=D^{0}F(X)+\sum_{i=1}^{n}Q_{t_{i}}u_{t_{i}}^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}},\cdots,X_{t_{n}}),$
(4.1)

where
$D^{0}F(X):=\sum_{i=1}^{n}\int_{0}^{t_{i}}Q_{t,t_{i}}u_{t_{i}}^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}},X_{t_{2}},\cdots,X_{t_{n}})\mathrm{d}t$
is the damped gradient on the path space with fixed initial point. Obviously,
$DF(X)\in L^{2}(\overline{\Omega}\rightarrow{\bf H};\overline{\mathbb{P}})$.
Define the Dirichlet form by

$\mathscr{E}^{\mu}(F,G)=\mathbb{E}_{\overline{\mathbb{P}}}\left<DF,DG\right>_{{\bf
H}},F,G\in\mathscr{F}C_{0}^{\infty}.$

We aim to prove that $(\mathscr{E}^{\mu},FC^{\infty}_{0})$ is closable in
$L^{2}(W^{T};\Pi^{T}_{\mu})$ and then establish the log-Sobolev inequality for
its closure $(\mathscr{E}^{\mu},\mathscr{D}(\mathscr{E}^{\mu}))$. To prove the
closability, we need the following two lemmas modified from [11]. Let
$\mathscr{H}_{0}(M)$ be the class of all smooth vector fields on $M$ with
compact support.

Let div${}^{0}_{\mu}$ be the divergence operator w.r.t. $\mu$, which is the
minus adjoint of $\nabla^{0}$ in $L^{2}(\mu)$; that is, for any smooth vector
field $U$,

$\displaystyle\int_{M}(Uf)\mathrm{d}\mu=-\int_{M}f({\rm
div}^{0}_{\mu}U)\mathrm{d}\mu,\ f\in C^{1}_{0}(M).$ (4.2)

######  Lemma 4.4.

Assume ${\bf(A)}$. For any $F\in\mathscr{F}C^{\infty}_{0}$,
$U\in\mathscr{H}_{0}(M)$, and $\overline{\mathscr{F}}_{t}$-adapted $h\in
L^{2}(\Omega\rightarrow{\bf H};\overline{\mathbb{P}})$,

$\mathbb{E}_{\overline{\mathbb{P}}}\left<DF(X),h+u_{0}^{-1}U(X_{0})\right>_{{\bf
H}}=\mathbb{E}_{\overline{\mathbb{P}}}\left\\{F(X)\left(\int_{0}^{T}\left<h^{\prime}(t),\mathrm{d}B_{t}\right>-({\rm
div}^{0}_{\mu}U)(X_{0})\right)\right\\}$

One can mimic the proof of [12], we omit it here. Due to this lemma, we have
the following result, the main idea is standard.

######  Theorem 4.5.

Assume ${\bf(A)}$. $(\mathscr{E}^{\mu},\mathscr{F}C^{\infty}_{0})$ is closable
in $L^{2}(W^{T};\Pi^{T}_{\mu})$, and its closure is symmetric Dirichlet form.

###### Proof.

It suffices to prove the closability. Let $\\{F_{n}\\}_{g\geq
1}\subset\mathscr{F}C_{0}^{\infty}$ such that
$\lim\limits_{n\rightarrow\infty}F_{n}=0$ in $L^{2}(W^{T};\Pi^{T}_{\mu})$ and
$V:=\lim\limits_{n\rightarrow\infty}DF_{n}(X)$ exists in
$L^{2}(\overline{\Omega}\rightarrow H;\overline{\mathbb{P}})$. We intend to
prove that $V=0$. Under the condition ${\bf(A)}$, by the decomposition of
identity, there is a smooth ONB $\\{U_{i}\\}_{i=1}^{d}$ for the tangent space.
Then for any $f\in C_{0}^{\infty}(M^{n})$, define

$\xi_{j}=\sum_{i=1}^{d}\left<u_{0}Q_{t_{i}}u_{t_{i}}^{-1}\nabla_{i}^{t_{i}}f(X_{t_{1}},X_{t_{2}},\cdots,X_{t_{n}}),U_{j}(X_{0})\right>_{0},\
\ j=1,2,\cdots,d,$

and there holds

$\displaystyle\sum_{i=1}^{n}Q_{t_{i}}u_{t_{i}}^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}},X_{t_{2}},\cdots,X_{t_{n}})=\sum_{i=1}^{\infty}\xi_{j}u_{0}^{-1}U_{j}(X_{0}).$
(4.3)

Combining this with (4.1), it suffices to prove
$\mathbb{E}_{\overline{\mathbb{P}}}\left<V,h+\xi
u_{0}^{-1}U(X_{0})\right>_{\bf H}=0$ for $\overline{\mathscr{F}}_{t}$-adapted
$h\in L^{2}(\overline{\Omega}\rightarrow{\bf H}_{0};\overline{\mathbb{P}}),\
\xi\in L^{2}(\overline{\Omega};\overline{\mathbb{P}})$ and
$U\in\mathscr{H}_{0}(M)$. Since $\mathscr{F}C_{0}^{\infty}$ is dense in
$L^{2}(W^{T};\Pi^{T}_{\mu})$, we may assume that $\xi=G(X)$ for some
$G\in\mathscr{F}C_{0}^{\infty}$. In this case, it follows from Lemma 4.4 and
Eq. (4.2) that

$\displaystyle\mathbb{E}_{\overline{\mathbb{P}}}\left<V,h+\xi
u_{0}^{-1}U(X_{0})\right>_{\bf
H}=\mathbb{E}_{\overline{\mathbb{P}}}\left<V(0),\xi
u_{0}^{-1}U(X_{0})\right>_{\bf
H}=\lim_{n\rightarrow\infty}\mathbb{E}_{\overline{\mathbb{P}}}\left<DF_{n}(X),G(X)u_{0}^{-1}U(X_{0})\right>_{\bf
H}$
$\displaystyle=\lim_{n\rightarrow\infty}\mathbb{E}_{\overline{\mathbb{P}}}\left\\{\left<\\{D(F_{n}G)(X)\\}(0),u_{0}^{-1}U(X_{0})\right>_{\bf
H}-F_{n}(X)\left<((DG)(X))(0),u_{0}^{-1}U(X_{0})\right>_{\bf H}\right\\}$
$\displaystyle=-\lim_{n\rightarrow\infty}\mathbb{E}_{\overline{\mathbb{P}}}\left\\{F_{n}(X)\left(G(X)({\rm
div}_{\mu}^{0}U)(X_{0})+\left<((DG)(X))(0),u_{0}^{-1}U(X_{0})\right>_{\bf
H}\right)\right\\}$ $\displaystyle=0.$

∎

By Theorems 4.3 and 4.5, we obtain the main result of this subsection as
follows.

######  Theorem 4.6.

Assume ${\bf(A)}$. If the log-Sobolev inequality

$\displaystyle\mu(f^{2}\log f^{2})\leq C\mu(|\nabla^{0}f|_{0}^{2}),\ f\in
C_{b}^{1}(M),\ \mu(f^{2})=1$ (4.4)

holds for some constant $C>0$, then

$\Pi^{T}_{\mu}(F^{2}\log F^{2})\leq(2\vee C)\mathscr{E}^{\mu}(F,F),\
F\in\mathscr{D}(\mathscr{E}^{\mu}),\ \Pi^{T}_{\mu}(F^{2})=1.$

###### Proof.

From Theorem 4.5, it suffice to prove $F\in\mathscr{F}C_{0}^{\infty}$. By
Theorem 4.3 and the condition (4.4), we have

$\displaystyle\Pi^{T}_{\mu}(F^{2}\log F^{2})$
$\displaystyle=\int_{M}\Pi_{x}^{T}(F^{2}\log F^{2})\mu(\mathrm{d}x)$
$\displaystyle\leq
2\int_{M}\mathscr{E}^{x}(F,F)\mu(\mathrm{d}x)+C\int_{M}\Pi_{x}^{T}(F^{2})\log\Pi_{x}^{T}(F^{2})\mu(\mathrm{d}x)$
$\displaystyle\leq 2\mathbb{E}_{\mathbb{P}}\|D^{0}F(X)\|^{2}_{{\bf
H}_{0}}+C\int_{M}|\nabla^{0}\sqrt{\mathbb{E}^{\cdot}F^{2}(X)}|_{0}^{2}\mathrm{d}\mu.$
(4.5)

Moreover, letting $F(X)=f(X_{t_{1}},X_{t_{2}},\cdots,X_{t_{n}})$, it follows
from Lemma 3.3 that

$\displaystyle|\nabla^{0}\sqrt{\mathbb{E}^{\cdot}F^{2}(X)}|_{0}^{2}$
$\displaystyle=\frac{|\mathbb{E}^{\cdot}F(X)\sum_{i=1}^{n}Q_{t_{i}}u_{t_{i}}^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}},\cdots,X_{t_{n}})|^{2}}{\mathbb{E}^{\cdot}F^{2}(X)}$
$\displaystyle\leq\mathbb{E}\big{|}\sum_{i=1}^{n}Q_{t_{i}}u_{t_{i}}^{-1}\nabla_{i}^{t_{i}}f(X_{t_{1}},\cdots,X_{t_{n}})\big{|}^{2}.$
(4.6)

Combining (4.2) with (4.2), we complete the proof. ∎

##  5 Transportation-cost inequalities on path spaces over convex manifolds

The main purpose of this section is to investigate the Talagrand type
inequalities on the path space $W^{S,T}:=C([S,T];M),\ 0\leq S<T<T_{c}$ of the
(reflecting) diffusion processes on the manifold with convex boundary under
$g_{t}$, $t\in[0,T_{c})$. Let $X_{t}$ be the (reflecting if $\partial
M\neq\varnothing$) diffusion process generated by $L_{t}$ with initial
distribution $\mu\in\mathscr{P}(M)$. Assume that $X_{t}$ is non-explosive,
which is the case if

$\mathcal{R}_{t}^{Z}\geq K(t),\,\mbox{for\ some}\ K\in C([0,T_{c})),\
\mathbb{I}_{t}\geq 0.$ (5.1)

We call the metric flow is convex flow if $(\partial M,g_{t})$ keeps convex,
i.e. $\mathbb{I}_{t}\geq 0$. Let $\Pi^{S,T}_{\mu}$ be the distribution of
$X_{[S,T]}:=\\{X_{t}:t\in[S,T]\\}$, $0\leq S<T<T_{c}$, which is a probability
measure on the (free) path space $W^{S,T}$. When $\mu=\delta_{x}$, we denote
$\Pi_{\delta_{x}}^{S,T}=\Pi_{x}^{S,T}$. For any nonnegative measurable
function $F$ on $W^{S,T}$ such that $\Pi^{S,T}_{\mu}(F)=1$, one has

$\displaystyle\mu_{F}^{S,T}(\mathrm{d}x):=\Pi^{S,T}_{x}(F)\mu(\mathrm{d}x)\in\mathscr{P}(M).$
(5.2)

Consider the uniform distance on $W^{S,T}$:

$\rho_{\infty}(\gamma,\eta):=\sup_{t\in[S,T]}\rho(\gamma_{t},\eta_{t}),\gamma,\eta\in
W^{S,T}.$

Let $W^{\rho_{\infty}}_{2}$ be the $L^{2}$-Wasserstein distance (or
$L^{2}$-transportation cost) induced by $\rho_{\infty}$. In general, for any
$p\in[1,\infty)$ and for two probability measures $\Pi_{1},\Pi_{2}$ on
$W^{S,T}$,

$W^{\rho_{\infty}}_{p}(\Pi_{1},\Pi_{2}):=\inf_{\pi\in\mathscr{C}(\Pi_{1},\Pi_{2})}\left\\{\int_{W^{S,T}\times
W^{S,T}}\rho_{\infty}(\gamma,\eta)^{p}\pi(\mathrm{d}\gamma,\mathrm{d}\eta)\right\\}^{1/p}$

is the $L^{p}$-Wasserstein distance (or $L^{p}$-transportation cost) of
$\Pi_{1}$ and $\Pi_{2}$, induced by the uniform norm, where
$\mathscr{C}(\Pi_{1},\Pi_{2})$ is the set of all couplings for $\Pi_{1}$ and
$\Pi_{2}$. Similarly the $L^{p}$-Wasserstein distance of $\mu$ and $\nu$
induced by $g_{t}$-distance defined by

$W_{p,t}(\nu,\mu)=\inf_{\eta\in\mathscr{C}(\nu,\mu)}\left\\{\int_{M\times
M}\rho_{t}(x,y)^{p}\mathrm{d}\eta(x,y)\right\\}^{1/p}.$

The following Theorem 5.2 provides that there are some transportation-cost
inequalities to be equivalent to the lower bound of $\mathcal{R}^{Z}_{t}$ and
the convexity of $(\partial M,g_{t})$, $t\in[0,T_{c})$ (when $\partial
M\neq\varnothing$). To prove this result, we need the following lemma due to
[OW].

######  Lemma 5.1.

Let $\mu$ be a probability measure on $M$ and $f\in C^{2}_{b}(M)$ such that
$\mu(f)=0$. For small enough $\varepsilon>0$ such that
$f_{\varepsilon}:=1+\varepsilon f\geq 0$, there holds

$\mu(f^{2})\leq\frac{1}{\varepsilon}\sqrt{\mu{(|\nabla^{s}f|_{s}^{2})}}W_{2,s}(f_{\varepsilon}\mu,\mu)+\frac{\|{\rm
Hess}^{s}_{f}\|_{\infty}}{2\varepsilon}W_{2,s}(f_{\varepsilon}\mu,\mu)^{2},$

where $\|{\rm Hess}^{s}_{f}\|_{\infty}=\sup_{x\in M}\|{\rm
Hess}^{s}_{f}\|_{op}$ for $\|\cdot\|_{op}$ the operator norm in
$\mathbb{R}^{d}$.

The main result of the section is presented as follows.

######  Theorem 5.2.

Let $P_{S,T}(x,\cdot)$ be the distribution of $X_{T}$ with conditional
$X_{S}=x$. Denote the corresponding inhomogeneous semigroup by
$\\{P_{S,T}\\}_{0\leq S\leq T<T_{c}}$. For any $p\in[1,\infty)$, the following
statements are equivalent to each other:

  1. $(1)$

(5.1) holds.

  2. $(2)$

For any $0\leq S\leq T<T_{c}$, $\mu\in\mathscr{P}(M)$ and nonnegative $F$ with
$\Pi^{S,T}_{\mu}(F)=1$,

$W_{2}^{\rho_{\infty}}(F\Pi^{S,T}_{\mu},\Pi^{S,T}_{\mu^{S,T}_{F}})\leq
4C(S,T,K)\Pi^{S,T}_{\mu}(F\log F)$

holds, where $\mu^{S,T}_{F}\in\mathscr{P}(M)$ is fixed by $(\ref{def2})$ and
$C(S,T,K):=\sup\limits_{t\in[S,T]}\int_{S}^{t}e^{-2\int_{u}^{t}K(r)\mathrm{d}r}\mathrm{d}u$,
which will keep the same meaning in $(3),(7)$ and $(8)$.

  3. $(3)$

For any $x\in M$ and $0\leq S\leq T<T_{c}$,

$W^{\rho_{\infty}}_{2}(F\Pi_{x}^{S,T},\Pi_{x}^{S,T})^{2}\leq
4C(S,T,K)\Pi_{x}^{S,T}(F\log F),\ F\geq 0,\ \Pi_{x}^{T}(F)=1.$

  4. $(4)$

For any $x\in M$ and $0\leq S\leq T<T_{c}$,

$W_{2,T}(P_{S,T}(x,\cdot),fP_{S,T}(x,\cdot))^{2}\leq
4\left(\int_{S}^{T}e^{-2\int_{u}^{T}K(r)\mathrm{d}r}\mathrm{d}u\right)P_{S,T}(f\log
f)(x),f\geq 0,\ P_{S,T}f(x)=1.$

  5. $(5)$

For any $x\in M$ and $0\leq S\leq T<T_{c}$,

$\displaystyle W_{2,T}(P_{S,T}(x,\cdot),fP_{S,T}(x,\cdot))^{2}\leq
4\left(\int_{S}^{T}e^{-2\int_{u}^{T}K(r)\mathrm{d}r}\mathrm{d}u\right)^{2}P_{S,T}$
$\displaystyle\frac{|\nabla^{T}f|_{T}^{2}}{f}(x),$

where $f\geq 1$ and $P_{S,T}f=1$.

  6. $(6)$

For $0\leq S\leq T<T_{c}$ and $\mu,\nu\in\mathscr{P}(M)$,

$W^{\rho_{\infty}}_{p}(\Pi^{S,T}_{\mu},\Pi_{\nu}^{S,T})\leq
e^{-\int_{S}^{T}K(r)\mathrm{d}r}W_{p,S}(\mu,\nu).$

  7. $(7)$

For any $0\leq S\leq T<T_{c}$, $\mu\in\mathscr{P}(M)$, and $F\geq 0$ with
$\Pi^{S,T}_{\mu}(F)=1$,

$W^{\rho_{\infty}}_{2}(F\Pi^{S,T}_{\mu},\Pi^{S,T}_{\mu})\leq
2\left\\{C(S,T,K)\Pi^{S,T}_{\mu}(F\log
F)\right\\}^{1/2}+e^{-\int_{S}^{T}K(r)\mathrm{d}r}W_{2,S}(\mu_{F}^{S,T},\mu).$

  8. $(8)$

For any $\mu\in\mathscr{P}(M)$ and $C\geq 0$ such that

$W_{2,S}(f\mu,\mu)^{2}\leq C\mu(f\log f),\ f\geq 0,\ \mu(f)=1,$

there holds

$\displaystyle
W^{\rho_{\infty}}_{2}(F\Pi^{S,T}_{\mu},\Pi^{S,T}_{\mu})\leq\left(2\sqrt{C(S,T,K)}+\sqrt{C}e^{-\int_{S}^{T}K(r)\mathrm{d}r}\right)^{2}\Pi^{S,T}_{\mu}(F\log
F),$

for $F\geq 0,\ \Pi^{S,T}_{\mu}(F)=1$.

###### Proof.

By taking $\mu=\delta_{x}$, we have
$\mu_{F}^{T}=\Pi^{T}_{x}(F)\delta_{x}=\delta_{x}$. It is easy to see that (3)
follows from (2), (7) and (8). (4) follows from (3) by taking
$F(X_{[S,T]})=f(X_{T})$. (6) implies

$W_{p,S}(\delta_{x}P_{S,T},\delta_{y}P_{S,T})\leq
e^{-\int_{S}^{T}K(r)\mathrm{d}r}\rho_{S}(x,y)$

and thus implies (1) by [7, Theorem 5.3]. Moreover, it is clear that (8)
follows from (7) while (7) is implied by each of (2) and (6).
“(3)$\Rightarrow$(2)” is the same as explained in time-homogeneous case (see
[24, the proof of Theorem 1.1 (b)]). So it suffices to prove (1) $\Rightarrow$
(3), each of (4) and (5) $\Rightarrow$ (1), (1) $\Rightarrow$ (5), (1)
$\Rightarrow$ (6). Without loss generality, we assume $S=0$ for simplicity.

${(a)}\ \mathbf{(1)\ implies\ (3)}$ We shall only consider the case where
$\partial M$ is non-empty. For the case without boundary, the following
argument works well by taking $l_{t}=0$ and $N_{t}=0$. Simply denote
$X^{x}_{[0,T]}=X_{[0,T]}$. Let $F$ be a positive bounded measurable function
on $W^{T}$ such that $\inf F>0$ and $\Pi^{T}_{x}(F)=1$. Let
$\mathrm{d}\mathbb{Q}=F(X_{[0,T]})\mathrm{d}\mathbb{P}$. Since
$\mathbb{E}F(X_{[0,T]})=\Pi^{T}_{\mu}(F)=1$, $\mathbb{Q}$ is a probability
measure on $\Omega$. Then, with a similar discussion as in [25], we conclude
that there exists a unique $\mathscr{F}_{t}$-predict process $\beta_{t}$ on
$\mathbb{R}^{d}$ such that

$F(X_{[0,T]})=m_{T}=e^{\int_{0}^{T}\left<\beta_{s},\mathrm{d}B_{s}\right>-\frac{1}{2}\int_{0}^{T}\|\beta_{s}\|^{2}\mathrm{d}s}$

and

$\displaystyle\int_{0}^{T}\mathbb{E}_{\mathbb{Q}}\|\beta_{s}\|^{2}\mathrm{d}s=2\mathbb{E}F(X_{[0,T]}\log
X_{[0,T]}).$ (5.3)

Then by the Girsanov theorem,
$\tilde{B}_{t}:=B_{t}-\int_{0}^{t}\beta_{s}\mathrm{d}s,\ \ t\in[0,T]$ is a
$d$-dimensional Brownian motion under the probability measure $\mathbb{Q}$.

Let $Y_{t}$ solve the following SDE

$\displaystyle\mathrm{d}Y_{t}=\sqrt{2}P^{t}_{X_{t},Y_{t}}u_{t}\circ\mathrm{d}\tilde{B}_{t}+Z_{t}(Y_{t})\mathrm{d}t+N_{t}(Y_{t})\mathrm{d}\tilde{l}_{t},\
\ Y_{0}=x,$ (5.4)

where $P^{t}_{X_{t},Y_{t}}$ is the $g_{t}$-parallel displacement along the
minimal geodesic from $X_{t}$ to $Y_{t}$ and $\tilde{l}_{t}$ is the local time
of $Y_{t}$ on $\partial M$. As announced, under $\mathbb{Q}$, $\tilde{B}_{t}$
is a $d$-dimensional Brownian motion, the distribution of $Y_{[0,T]}$ is
$\Pi_{x}^{T}$.

On the other hand, since
$\tilde{B}_{t}=B_{t}-\int_{0}^{t}\beta_{s}\mathrm{d}s$, we have

$\displaystyle\mathrm{d}X_{t}=\sqrt{2}u_{t}\circ\mathrm{d}\tilde{B}_{t}+Z_{t}(X_{t})\mathrm{d}t+\sqrt{2}u_{t}\beta_{t}\mathrm{d}t+N_{t}(X_{t})\mathrm{d}l_{t}.$
(5.5)

Moreover, for any bounded measurable function $G$ on $W^{T}$,

$\mathbb{E}_{\mathbb{Q}}G(X_{[0,T]}):=\mathbb{E}(FG)(X_{[0,T]})=\Pi_{x}^{T}(FG).$

We conclude that the distribution of $X_{[0,T]}$ under $\mathbb{Q}$ coincides
with $F\Pi^{T}_{x}$. Therefore,

$\displaystyle W^{\rho_{\infty}}_{2}(F\Pi_{x}^{T},\Pi_{x}^{T})^{2}$
$\displaystyle\leq\mathbb{E}_{\mathbb{Q}}\rho_{\infty}(X_{[0,T]},Y_{[0,T]})^{2}=\mathbb{E}_{\mathbb{Q}}\max_{t\in[0,T]}\rho_{t}(X_{t},Y_{t})^{2}.$
(5.6)

By the convexity of $(\partial M,g_{t})$, we have

$\left<N_{t}(x),\nabla^{t}\rho_{t}(\cdot,y)(x)\right>_{t}=\left<N_{t}(x),\nabla^{t}\rho_{t}(y,\cdot)(x)\right>_{t}\leq
0.$

Combining this with (5.4) and (5.5), and by the Itô formula, we obtain from
$\mathcal{R}_{t}^{Z}\geq K(t)$ that

$\displaystyle\mathrm{d}\rho_{t}(X_{t},Y_{t})\leq$
$\displaystyle-K(t)\rho_{t}(X_{t},Y_{t})\mathrm{d}t+\sqrt{2}\left<u_{t}\beta_{t},\nabla^{t}\rho_{t}(\cdot,Y_{t})(X_{t})\right>_{t}\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle(-K(t)\rho_{t}(X_{t},Y_{t})+\sqrt{2}\|\beta_{t}\|)\mathrm{d}t.$
(5.7)

Since $X_{0}=Y_{0}=x$, this implies

$\displaystyle\rho_{t}(X_{t},Y_{t})^{2}$ $\displaystyle\leq
e^{-2\int_{0}^{t}K(r)\mathrm{d}r}\left(\sqrt{2}\int_{0}^{t}e^{\int_{0}^{s}K(r)\mathrm{d}r}\|\beta_{s}\|\mathrm{d}s\right)^{2}$
$\displaystyle\leq
2e^{-2\int_{0}^{t}K(r)\mathrm{d}r}\int_{0}^{t}e^{2\int_{0}^{s}K(r)\mathrm{d}r}\mathrm{d}s\cdot\int_{0}^{t}\|\beta_{s}\|^{2}\mathrm{d}s,\
\ t\in[0,T].$ (5.8)

Therefore,

$\displaystyle\mathbb{E}_{\mathbb{Q}}\max_{t\in[0,T]}\rho_{t}(X_{t},Y_{t})^{2}\leq
2\max_{t\in[0,T]}\int_{0}^{t}e^{-2\int_{s}^{t}K(r)\mathrm{d}r}\mathrm{d}s\int_{0}^{t}\mathbb{E}_{\mathbb{Q}}\|\beta_{s}\|^{2}\mathrm{d}s.$
(5.9)

Therefore, (3) follows from (5.6) and (5.3).

${(b)}\ \mathbf{(4)\ implies\ (1)\ }$ Let $f\in C_{b}^{2}(M)$ such that
$P_{0,T}f(x)=0$. Then, for small $\varepsilon>0$ such that
$f_{\varepsilon}:=1+\varepsilon f\geq 0$, we have

$\displaystyle P_{0,T}(f_{\varepsilon}\log
f_{\varepsilon})=P_{0,T}\left\\{(1+\varepsilon f)\left(\varepsilon
f-\frac{1}{2}(\varepsilon
f)^{2}+o(\varepsilon^{2})\right)\right\\}(x)=\frac{\varepsilon^{2}}{2}P_{0,T}f^{2}(x)+o(\varepsilon^{2}).$

Combining with lemma 5.1 and (4), we obtain

$\displaystyle(P_{0,T}f^{2})^{2}(x)$ $\displaystyle\leq
4\int_{0}^{T}e^{-2\int_{u}^{T}K(r)\mathrm{d}r}\mathrm{d}u\cdot
P_{0,T}|\nabla^{T}f|^{2}_{T}(x)\cdot\lim_{\varepsilon\rightarrow
0}\frac{P_{0,T}f_{\varepsilon}\log f_{\varepsilon}(x)}{\varepsilon^{2}}$
$\displaystyle 4\int_{0}^{T}e^{-2\int_{u}^{T}K(r)\mathrm{d}r}\mathrm{d}u\cdot
P_{0,T}|\nabla^{T}f|^{2}_{T}(x)P_{0,T}f^{2}(x).$

This is equivalent to [7, Theorem 5.3] for $\sigma=0,p=2$ and continuous
function $K$. Therefore,by [7, Theorem 5.3] “(2)$\Leftrightarrow$ (1)”, (4)
implies (1).

${(c)}\mathbf{(5)\ is\ equivalent\ to\ (1)}$. Similarly to $\mathbf{(b)}$,
combining condition (5) with Lemma 5.1, we obtain

$\displaystyle P_{0,T}f^{2}(x)$ $\displaystyle\leq
2\int_{0}^{T}e^{-2\int_{s}^{T}K(r)\mathrm{d}r}\mathrm{d}s\sqrt{P_{0,T}|\nabla^{T}f|^{2}_{T}(x)}\lim_{\varepsilon\rightarrow
0}\sqrt{P_{0,T}\frac{|\nabla^{T}f_{\varepsilon}|^{2}_{T}}{f_{\varepsilon}\varepsilon^{2}}(x)}$
$\displaystyle=2\int_{0}^{T}e^{-2\int_{s}^{T}K(r)\mathrm{d}r}\mathrm{d}sP_{0,T}|\nabla^{T}f|^{2}_{T}.$

Hence, (1) holds.

On the other hand, due to $(1)\Rightarrow(4)$ and (5.1),

$\displaystyle P_{0,T}(f\log
f)(x)\leq\int_{0}^{T}e^{-2\int_{s}^{T}K(r)\mathrm{d}r}\mathrm{d}s\cdot
P_{0,T}\frac{|\nabla^{T}f|_{T}^{2}}{f}(x),\ \ f\geq 0,\ \ P_{0,T}f(x)=1,$

we conclude that (1) implies (5).

${(e)}\mathbf{(1)\ implies\ (2)}$. For any $x,y\in M$, there exists
$\Pi_{x,y}\in\mathscr{C}(\Pi_{x}^{T},\Pi_{y}^{T})$ such that

$\int_{W^{T}\times W^{T}}\rho_{\infty}^{p}\mathrm{d}\Pi_{x,y}\leq
e^{-p\int_{0}^{T}K(r)\mathrm{d}r}\rho_{0}(x,y)^{p}.$

The following discussion is similar with the constant matric case (see [24,
Theorem 1.1]). ∎

##  6 Extension to non-convex setting

In this section, we first consider $L_{t}=\psi^{2}_{t}(\nabla^{t}+Z_{t})$ with
diffusion coefficient $\psi_{t}$ on manifolds with convex flow; then extend
these results to non-convex case.

###  6.1 The case with a diffusion coefficient

Let $\psi_{t}(\cdot)=\psi(t,\cdot)>0$ be a smooth function on $(M,g_{t})$ and
$\Pi^{T}_{\mu,\psi}$ be the distribution of the (reflecting if $\partial
M\neq\varnothing$) diffusion process generated by
$L_{t}=\psi_{t}^{2}(\nabla^{t}+Z_{t})$ on time interval
$[0,T]\subset[0,T_{c})$ with initial distribution $\mu$. Set
$\Pi^{T}_{x,\psi}=\Pi^{T}_{\delta_{x},\psi}$ for $x\in M$. Moreover, for
$F\geq 0$ with $\Pi^{T}_{\mu,\psi}=0$, let
$\mu^{T}_{F,\psi}(\mathrm{d}x)=\Pi^{T}_{s,\psi}(F)\mu(\mathrm{d}x).$

######  Theorem 6.1.

Assume that $\mathbb{I}_{t}\geq 0$ for $t\in[0,T_{c})$ and
$\mathrm{Ric}_{t}^{Z}\geq K_{1}(t),\ \mathcal{G}_{t}\leq K_{2}(t)$ for some
continuous function on $[0,T_{c})$. Let $\psi\in
C^{1,\infty}_{b}([0,T_{c})\times M)$ be strictly positive. Let

$K_{\psi}(t)=(d-1)\|\nabla^{t}\psi_{t}\|_{\infty}^{2}+K_{1}^{-}(t)\|\psi_{t}\|_{\infty}^{2}+2\|Z_{t}\|_{\infty}\|\psi_{t}\|_{\infty}\|\nabla^{t}\psi_{t}\|_{\infty}+K_{2}(t).$

Then

$W^{\rho_{\infty}}_{2}(F\Pi^{T}_{\mu,\psi},\Pi^{T}_{\mu^{T}_{F,\psi},\psi})^{2}\leq
C(T,\psi)\Pi^{T}_{\mu,\psi}(F\log F),\ \mu\in\mathscr{P}(M),\ F\geq 0,\
\Pi^{T}_{\mu,\psi}(F)=1$

holds for

$C(T,\psi):=\inf_{R>0}\left\\{4(1+R^{-1})\int_{0}^{T}\|\psi_{s}\|_{\infty}^{2}e^{2\int_{s}^{T}K_{\psi}(r)\mathrm{d}r}\mathrm{d}s\cdot\exp\bigg{[}8(1+R)\sup_{s\in[0,T]}\|\nabla^{s}\psi_{s}\|_{\infty}\bigg{]}\right\\}.$

###### Proof.

We shall only consider the case that $\partial M$ is non-empty. As explained
in the proof of “(3)$\Rightarrow$(2)” in [24, Theorem 4.1], it suffices to
prove for $\mu=\delta_{x}$, $x\in M$. In this case, the desired inequality
reduces to

$W_{2}^{\rho_{\infty}}(F\Pi_{x,\psi}^{T},\Pi_{x,\psi}^{T})\leq
2C(T,\psi)\Pi_{x,\psi}^{T}(F\log F),\ \ F\geq 0,\ \Pi_{x,\psi}^{T}(F)=1.$

Since the diffusion coefficient is non-constant, it is convenient to adopt the
Itô differential $\mathrm{d}_{I}$ for the Girsanov transformation. So the
$L_{t}$-reflecting diffusion process can be constructed by solving the Itô SDE

$\mathrm{d}_{I}X_{t}=\sqrt{2}\psi_{t}(X_{t})u_{t}\mathrm{d}B_{t}+\psi_{t}^{2}(X_{t})Z_{t}(X_{t})\mathrm{d}t+N_{t}(X_{t})\mathrm{d}l_{t},\
\ \ X_{0}=x,$

where $B_{t}$ is the $d$-dimensional Brownian motion with natural filtration
$\mathscr{F}_{t}$. Let $\beta_{t}$, $\mathbb{Q}$ and $\tilde{B}_{t}$ be the
same as in the proof of Theorem 5.2. Then

$\displaystyle\mathrm{d}_{I}X_{t}=\sqrt{2}\psi_{t}(X_{t})u_{t}\mathrm{d}\tilde{B}_{t}+\\{\psi_{t}^{2}(X_{t})Z_{t}(X_{t})+\sqrt{2}\psi_{t}(X_{t})u_{t}\beta_{t}\\}\mathrm{d}t+N_{t}(X_{t})\mathrm{d}l_{t}.$
(6.1)

Let $Y_{t}$ solve

$\displaystyle\mathrm{d}_{I}Y_{t}=\sqrt{2}\psi_{t}(Y_{t})P^{t}_{X_{t},Y_{t}}u_{t}\mathrm{d}\tilde{B}_{t}+\psi_{t}^{2}(Y_{t})Z_{t}(Y_{t})\mathrm{d}t+N_{t}(Y_{t})\mathrm{d}\tilde{l}_{t},\
\ Y_{0}=y,$ (6.2)

where $\tilde{l}_{t}$ is the local time of $Y_{t}$ on $\partial M$. As
explained in the above theorem (see e.g. the proof of Theorem 5.2 (a)), under
$\mathbb{Q}$, the distribution of $Y_{[0,T]}$ and $X_{[0,T]}$ are
$\Pi_{x,\psi}^{T}$ and $F\Pi_{x,\psi}^{T}$, so

$\displaystyle
W_{2}^{\rho_{\infty}}(F\Pi_{x,\psi}^{T},\Pi_{x,\psi}^{T})\leq\mathbb{E}_{\mathbb{Q}}\max_{t\in[0,T]}\rho_{t}(X_{t},Y_{t})^{2}.$
(6.3)

Noting that due to the convexity of the boundary,

$\left<N_{t}(x),\nabla^{t}\rho(\cdot,y)(x)\right>_{t}=\left<N_{t}(y),\nabla^{t}\rho_{t}(y,\cdot)(x)\right>_{t}\leq
0,\ \ x\in\partial M,$

Combining this with (6.1), (6.2) and the comparison theorem [7, Theorem 4.1],
we obtain

$\displaystyle\mathrm{d}\rho_{t}(X_{t},Y_{t})\leq$
$\displaystyle\sqrt{2}(\psi_{t}(X_{t})-\psi_{t}(Y_{t}))\left<\nabla^{t}\rho_{t}(\cdot,Y_{t})(X_{t}),u_{t}\mathrm{d}B_{t}\right>_{t}$
$\displaystyle+K_{\psi}(t)\rho_{t}(X_{t},Y_{t})\mathrm{d}t+\sqrt{2}\|\psi_{t}\|_{\infty}\|\beta_{t}\|\mathrm{d}t,$

where

$K_{\psi}(t)=(d-1)\|\nabla^{t}\psi_{t}\|_{\infty}^{2}+K^{-}_{1}(t)\|\psi_{t}\|^{2}_{\infty}+2\|Z_{t}\|_{\infty}\|\nabla^{t}\psi_{t}\|_{\infty}\|\psi_{t}\|_{\infty}+K_{2}(t).$

Then

$M_{t}:=\sqrt{2}\int_{0}^{t}e^{-\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}(\psi_{s}(X_{s})-\psi_{s}(Y_{s}))\left<\nabla^{s}\rho_{s}(\cdot,Y_{s})(X_{s}),u_{s}\mathrm{d}\tilde{B}_{s}\right>_{s}$

is a $\mathbb{Q}$-martingale such that

$\displaystyle\rho_{t}(X_{t},Y_{t})\leq
e^{\int_{0}^{t}K_{\psi}(r)\mathrm{d}r}\left(M_{t}+\sqrt{2}\int_{0}^{t}e^{-\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}\|\psi_{s}\|_{\infty}\|\beta_{s}\|\mathrm{d}s\right),\
\ t\in[0,T].$

So by the Doob inequality, we obtain

$\displaystyle h_{t}:=$ $\displaystyle
e^{-2\int_{0}^{t}K_{\psi}(s)\mathrm{d}s}\mathbb{E}\max_{s\in[0,t]}\rho_{s}(X_{s},Y_{s})^{2}$
$\displaystyle\leq$
$\displaystyle(1+R)\mathbb{E}_{\mathbb{Q}}\max_{s\in[0,t]}M_{s}^{2}+2(1+R^{-1})\mathbb{E}_{\mathbb{Q}}\left(\int_{0}^{t}e^{-\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}\|\psi_{s}\|_{\infty}\|\beta_{s}\|\mathrm{d}s\right)^{2}$
$\displaystyle\leq$ $\displaystyle
4(1+R)\mathbb{E}_{\mathbb{Q}}M_{t}^{2}+2(1+R^{-1})\int_{0}^{t}e^{-2\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}\|\psi_{s}\|_{\infty}^{2}\mathrm{d}s\int_{0}^{t}\mathbb{E}_{\mathbb{Q}}\|\beta_{s}\|^{2}\mathrm{d}s$
$\displaystyle\leq$ $\displaystyle
8(1+R)\sup_{s\in[0,T]}\|\nabla^{s}\psi_{s}\|_{\infty}\int_{0}^{t}h_{s}\mathrm{d}s$
$\displaystyle+2(1+R^{-1})\int_{0}^{T}\|\psi_{s}\|_{\infty}^{2}e^{-2\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}\mathrm{d}s\cdot\int_{0}^{T}\mathbb{E}\|\beta_{s}\|^{2}\mathrm{d}s,\
\ \ t\in[0,T].$

Since $h_{0}=0$, this inequality implies

$\displaystyle
e^{-2\int_{0}^{T}K_{\psi}(s)\mathrm{d}s}\mathbb{E}_{\mathbb{Q}}\max_{s\in[0,T]}\rho_{s}(X_{s},Y_{s})^{2}=h_{T}$
$\displaystyle\leq
2(1+R^{-1})\int_{0}^{T}\|\psi_{s}\|_{\infty}^{2}e^{-2\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}\mathrm{d}s\cdot\exp\left[8(1+R)\sup_{s\in[0,T]}\|\nabla^{s}\psi_{s}\|_{\infty}\right]\cdot\int_{0}^{T}\mathbb{E}_{\mathbb{Q}}\|\beta_{s}\|^{2}\mathrm{d}s.$

As explained in (5.3), it holds

$\int_{0}^{T}\mathbb{E}_{\mathbb{Q}}\|\beta_{s}\|^{2}\mathrm{d}s=2\mathbb{E}F(X_{[0,T]})\log
F(X_{[0,T]}).$

Therefore

$\displaystyle\mathbb{E}\max_{s\in[0,T]}\rho_{s}(X_{s},Y_{s})^{2}$
$\displaystyle\leq
4(1+R^{-1})\int_{0}^{T}\|\psi_{s}\|_{\infty}^{2}e^{2\int_{s}^{T}K_{\psi}(r)\mathrm{d}r}\mathrm{d}s\cdot\exp{\big{[}8(1+R)\sup_{s\in[0,T]}\|\nabla^{s}\psi_{s}\|_{\infty}\big{]}\Pi_{o,\psi}^{T}(F\log
F)}.$

Combining with (6.3), we complete the proof. ∎

######  Theorem 6.2.

In the situation of Theorem 6.1,

$W^{\rho_{\infty}}_{2}(\Pi^{T}_{\nu,\psi},\Pi^{T}_{\mu,\psi})\leq
2e^{\int_{0}^{T}(K_{\psi}(t)+\|\nabla^{t}\psi_{t}\|_{\infty})\mathrm{d}t}W_{2,0}(\nu,\mu).$

###### Proof.

As explained in the proof of Theorem 5.2 $``(6)\Rightarrow(5)"$, we only
consider $\nu=\delta_{x}$, and $\nu=\delta_{y}$. Let $X_{t}$ and $Y_{t}$ solve
the following SDEs respectively.

$\displaystyle\mathrm{d}_{I}X_{t}=\sqrt{2}\psi_{t}(X_{t})u_{t}\mathrm{d}B_{t}+\psi_{t}^{2}(X_{t})Z_{t}(X_{t})\mathrm{d}t+N_{t}(X_{t})\mathrm{d}l_{t},\
\ X_{0}=x;$
$\displaystyle\mathrm{d}_{I}Y_{t}=\sqrt{2}\psi_{t}(Y_{t})P_{X_{t},Y_{t}}^{t}u_{t}\mathrm{d}B_{t}+\psi_{t}^{2}(Y_{t})Z_{t}(Y_{t})\mathrm{d}t+N_{t}(Y_{t})\mathrm{d}\tilde{l_{t}},\
\ Y_{0}=y.$

Then, as explained in Theorem 6.1, by the Itô formula,

$\displaystyle\mathrm{d}\rho_{t}(X_{t},Y_{t})$
$\displaystyle\leq\sqrt{2}(\psi_{t}(X_{t})-\psi_{t}(Y_{t}))\left<\nabla^{t}\rho_{t}(\cdot,Y_{t})(X_{t}),u_{t}\mathrm{d}B_{t}\right>_{t}+K_{\psi}(t)\rho_{t}(X_{t},Y_{t})\mathrm{d}t.$
(6.4)

Therefore,

$\displaystyle\rho_{t}(X_{t},Y_{t})\leq
e^{\int_{0}^{t}K_{\psi}(s)\mathrm{d}s}(M_{t}+\rho_{0}(x,y)),\ t\geq 0,$ (6.5)

for
$M_{t}:=\sqrt{2}\int_{0}^{t}e^{-\int_{0}^{s}K_{\psi}(u)\mathrm{d}u}(\psi_{s}(X_{s})-\psi_{s}(Y_{s}))\left<\nabla^{s}\rho_{s}(\cdot,Y_{s})(X_{s}),u_{s}\mathrm{d}B_{s}\right>_{s}.$
Again using the Itô formula,

$\displaystyle\mathrm{d}\rho_{t}^{2}(X_{t},Y_{t})\leq\mathrm{d}\tilde{M}_{t}+2\left[K_{\psi}(t)+\|\nabla^{t}\psi_{t}\|_{\infty}^{2}\right]\rho_{t}(X_{t},Y_{t})^{2}\mathrm{d}t,$

where
$\mathrm{d}\tilde{M}_{t}=2\rho_{t}(X_{t},Y_{t})(\psi_{t}(X_{t})-\psi_{t}(Y_{t}))\left<\nabla^{t}\rho_{t}(\cdot,Y_{t})(X_{t}),u_{t}\mathrm{d}B_{t}\right>.$
This implies

$\displaystyle\mathrm{d}\rho_{t}^{2}(X_{t},Y_{t})\leq
e^{2\int_{0}^{t}(K_{\psi}(s)+\|\nabla^{s}\psi_{s}\|_{\infty})\mathrm{d}s}\rho_{0}(x,y)^{2}.$

Combining this with (6.5), we arrive at

$\displaystyle W_{2}^{\rho_{\infty}}(\Pi^{T}_{x,\psi},\Pi^{T}_{y,\psi})^{2}$
$\displaystyle\leq\mathbb{E}\max_{t\in[0,T]}\rho_{t}(X_{t},Y_{t})^{2}\leq
e^{2\int_{0}^{T}K_{\psi}(t)\mathrm{d}t}\mathbb{E}\max_{t\in[0,T]}(M_{t}+\rho_{0}(x,y))^{2}$
$\displaystyle\leq
4e^{2\int_{0}^{T}K_{\psi}(t)\mathrm{d}t}\mathbb{E}(M_{T}+\rho_{0}(x,y))^{2}=4e^{2\int_{0}^{T}K_{\psi}(t)\mathrm{d}t}\mathbb{E}(M_{T}^{2}+\rho_{0}^{2}(x,y))$
$\displaystyle\leq
4e^{2\int_{0}^{T}K_{\psi}(t)\mathrm{d}t}\left(\rho_{0}(x,y)^{2}+2\int_{0}^{T}e^{-2\int_{0}^{t}K_{\psi}(s)\mathrm{d}s}\|\nabla^{t}\psi_{t}\|_{\infty}\mathbb{E}\rho_{t}(X_{t},Y_{t})^{2}\mathrm{d}t\right)$
$\displaystyle\leq
4e^{2\int_{0}^{T}(K_{\psi}(t)+\|\nabla^{t}\psi_{t}\|_{\infty})\mathrm{d}t}\rho_{0}(x,y)^{2}$

where the second inequality is due to the Doob inequality. This implies the
desired inequality for $\mu=\delta_{x}$ and $\nu=\delta_{y}$. ∎

###  6.2 Non-convex manifold

As discussed in Theorem 6.1 and with a proper conformal change of metric, we
are able to establish the following transportation-cost inequality on a class
of manifolds with non-convex boundary. Let

$\mathscr{D}=\\{\phi\in C_{b}^{2}([0,T_{c})\times M):\inf\phi_{t}=1,\
\mathbb{I}_{t}\geq-N_{t}\log\phi_{t}\\}.$

Assume that $\mathscr{D}\neq\varnothing$ and for some $K_{1},K_{2}\in
C{([0,T_{c}))}$ such that

$\displaystyle{\rm Ric}_{t}^{Z}\geq K_{1}(t),\ \ \mathcal{G}_{t}\leq K_{2}(t)$
(6.6)

holds. To make the boundary convex, let $\phi_{t}\in\mathscr{D}$. By Theorem
[23, Lemma 2.1], $\partial M$ become convex under
$\tilde{g}_{t}=\phi_{t}^{-2}g_{t}$. Let $\tilde{\Delta}_{t}$ and
$\tilde{\nabla}^{t}$ be the Laplacian and gradient induced by the new metric
$\tilde{g}_{t}$. Since $\phi_{t}\geq 1$, $\rho_{t}(x,y)$ is large than
$\tilde{\rho}_{t}(x,y)$, the Riemannian $\tilde{g}_{t}$-distance between $x$
and $y$.

######  Theorem 6.3.

Let $\partial M\neq\varnothing$ and $\mathbb{I}_{t}\geq-\sigma(t)$ for
positive $\sigma\in C([0,T_{c}))$. Assume (6.6) holds. For
$\phi\in\mathscr{D}$, let

$K_{\phi}(t):=(d-1)\|\nabla^{t}\phi_{t}\|_{\infty}^{2}+K_{\phi,1}^{-}(t)+2\|\phi_{t}Z_{t}+(d-2)\nabla^{t}\phi_{t}\|_{\infty}\|\nabla^{t}\phi_{t}\|_{\infty}+K_{\phi,2}(t),$

where

$\displaystyle
K_{\phi,1}(t):=\inf\\{\phi_{t}K_{1}(t)+\frac{1}{2}L_{t}\phi_{t}^{2}-|\nabla^{t}\phi_{t}^{2}|_{t}|Z_{t}|_{t}-(d-2)|\nabla^{t}\phi_{t}|_{t}^{2}\\},$
$\displaystyle K_{\phi,2}(t):=\sup\\{-2\partial_{t}\log\phi_{t}+K_{2}(t)\\}.$

Then for any $\mu\in\mathscr{P}(M)$,

$W^{\rho_{\infty}}_{2}(F\Pi_{\mu}^{T},\Pi_{\mu_{F}^{T}}^{T})\leq\sup_{s\in[0,T]}\|\phi_{t}\|_{\infty}^{2}C(T,\phi)\Pi_{\mu}^{T}(F\log
F),\ \ F\geq 0,\ \Pi_{\mu}^{T}(F)=1$

holds for

$C(T,\phi)=\inf_{R>0}\left\\{4(1+R^{-1})\int_{0}^{T}e^{2\int_{s}^{T}K_{\phi}(r)\mathrm{d}r}\mathrm{d}s\exp{\bigg{[}8(1+R)\sup_{s\in[0,T]}\|\nabla^{s}\phi_{s}\|_{\infty}\bigg{]}}\right\\}.$

###### Proof.

According to the proof of [7, Proposition 4.7]. We have
$L_{t}:=\phi_{t}^{-2}(\tilde{\Delta}_{t}+\tilde{Z}_{t})$, where
$\tilde{Z}_{t}=\phi_{t}^{2}Z_{t}+\frac{d-2}{2}\nabla^{t}\phi_{t}^{2}$, and

$\widetilde{{\rm Ric}}^{\tilde{Z}}_{t}\geq K_{\phi,1}(t),\ \ \
\tilde{\mathcal{G}}_{t}\leq K_{\phi,2}(t).$

Let $K_{\psi}$ be defined in Theorem 6.3 for the manifold equipped with
$\tilde{g}_{t}$. Then, $L_{t}=\psi_{t}^{2}(\tilde{\Delta}_{t}+\tilde{Z}_{t})$,
where $\psi_{t}=\phi_{t}^{-1}$, we see that $K_{\psi}(t)\leq K_{\phi}(t)$ and
thus $C(t,\psi)\leq C(t,\phi)$. Hence, it follows from Theorem 6.3 that

$W^{\tilde{\rho}_{\infty}}_{2}(F\Pi_{\mu}^{T},\Pi^{T}_{\mu^{T}_{F}})^{2}\leq
C(T,\phi)\Pi_{\mu}^{T}(F\log F),\ \ F\geq 0,\ \ \Pi_{\mu}^{T}(F)=1,$

where $\tilde{\rho}_{\infty}$ is the uniform distance on $W^{T}$ induced by
the metric $\tilde{g}_{t}$. The proof is completed by
$\rho_{\infty}\leq\sup_{t\in[0,T]}\|\phi_{t}\|_{\infty}\tilde{\rho}_{\infty}.$
∎

Since $K_{\psi}(t)\leq K_{\phi}(t)$ and
$\tilde{\rho}_{t}\leq\rho_{t}\leq\|\phi_{t}\|_{\infty}\tilde{\rho}_{t}$, the
following result follows from the proof of Theorem 3.2 by taking
$\psi=\phi^{-1}$.

######  Theorem 6.4.

In the situation of Theorem 6.3,

$W^{\rho_{\infty}}_{2}(\Pi^{T}_{\mu},\Pi_{\nu}^{T})\leq
2\sup_{t\in[0,T]}\|\phi_{t}\|_{\infty}e^{\int_{0}^{T}(K_{\phi}(t)+\|\nabla^{t}\phi_{t}\|_{\infty})\mathrm{d}t}W_{2,0}(\nu,\mu),\
\ \mu,\nu\in\mathscr{P},T>0.$

Acknowledgements The author would thank Professor Feng-Yu Wang for valuable
suggestions and this work is supported in part by 985 Project, 973 Project.

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