Transportation-Cost Inequalities

on Path Space over Manifolds with Boundary^{1}

Feng-Yu Wang^{1}

Received: May 9, 2011 Revised: October 11, 2011

Communicated by Friedrich Götze

Abstract. Let L = Δ + Z for a C^{1} vector field Z on a complete Riemannian manifold possibly with a boundary. A number of transportation-cost inequalities on the path space for the (reflecting) L-diffusion process are proved to be equivalent to the curvature condition Ric − ∇Z ≥ −K and the convexity of the boundary (if exists). These inequalities are new even for manifolds without boundary, and are partly extended to non-convex manifolds by using a conformal change of metric which makes the boundary from non-convex to convex.

2010 Mathematics Subject Classification: 60J60, 58G60. Keywords and Phrases: Transportation-cost inequality, curvature, second fundamental form, path space.

1 Introduction

In 1996 Talagrand [13] found that the L^{2}-Wasserstein distance to the standard Guassian measure can be dominated by the square root of twice relative entropy. This inequality is called (Talagrand) transportation-cost inequality, and 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 e.g. [7] for the study on the Wiener space with the Cameron-Martin distance, [17, 5] on the path space of diffusions with the L^{2}-distance, [18] on the Riemannian path space with intrinsic distance induced by the Malliavin gradient operator, and [6, 27] on the path space of diffusions with the uniform distance. The main purpose of this paper is to investigate the Talagrand inequality on the path space of reflecting diffusion process, for which both the curvature and the second fundamental form of the boundary will take important roles. Let M be a connected complete Riemannian manifold possibly with a boundary ∂M. Let L = Δ + Z for a C^{1} vector field Z on M. Let X_{t} be the (reflecting if ∂M ≠ ∅) diffusion process generated by L with initial distribution μ ∈ 𝒫(M), where 𝒫(M) is the set of all probability measures on M. Assume that X_{t} is non-explosive, which is the case if ∂M is convex and the curvature condition

1Supported in Part By NNSFC(11131003), SRFDP and the Fundamental Research Funds for the Central Universities.

Documenta Mathematica 18 (2013) 297–322

Ric − ∇Z ≥ −K  (1.1)

holds for some constant K ∈ ℝ. In this case, for any T > 0, the distribution Π_{μ}^{T} of X_{[0,T]} := {X_{t} : t ∈ [0, T]} is a probability measure on the (free) path space

M^{T} := C([0, T]; M).

When μ = δ_{o}, the Dirac measure at point o ∈ M, we simply denote Π_{δ_{o}}^{T} = Π_{o}^{T} . For any nonnegative measurable function F on M_{T} such that Π_{μ}^{T} (F) = 1, one has

μ_{F}^{T} (dx) := Π_{x}^{T} (F)μ(dx) ∈ 𝒫(M).  (1.2)

Let ρ be the Riemannian distance on M; i.e. for x, y ∈ M, ρ(x, y) is the length of the shortest curve on M linking x and y. Then M^{T} is a Polish space under the uniform distance

ρ_{∞}(γ, η) = sup_{t∈[0,T]} ρ(γ_{t}, η_{t}), γ, η ∈ M^{T} .

Let W_{2,ρ_{∞}} be the L^{2}-Wasserstein distance (or L^{2}-transportation cost) induced by ρ_{∞}. In general, for any p ≥ 1 and for two probability measures Π_{1}, Π_{2} on M^{T},

W_{p,ρ_{∞}}(Π_{1}, Π_{2}) := inf_{π∈𝒞 (Π_{1},Π_{2})} { ∫_{M_{T} ×M_{T}} ρ_{∞}(γ, η)^{p}π(dγ, dη)}^{1/p}

is the L^{p}-Warsserstein distance (or L^{p}-transportation cost) of Π_{1} and Π_{2} induced by the uniform norm, where 𝒞 (Π_{1}, Π_{2}) is the set of all couplings for Π_{1} and Π_{2}. Before moving on, let us recall the Talagrand transportation-cost inequality established in [6] on the path space over Riemannian manifolds without boundary. Let ∂M = ∅ and ρ_{o} = ρ(o, ·). If

|Z| ≤ ψ ∘ ρ_{o}  (1.3)

holds for some positive function ψ such that ∫_{0}^{∞} \frac{1}{ψ(s)}ds = ∞, then (see [6, Theorem 1.1])

W_{2,ρ_{∞}}(F Π_{o}^{T} , Π_{o}^{T})^{2} ≤ \frac{2}{K} (e^{2KT} − 1)Π_{o}^{T} (F log F), F ≥ 0, Π_{o}^{T} (F) = 1. (1.4)

According to [12, 4, 18], the log-Sobolev inequality for a smooth elliptic diffusion implies the Talagrand transportation-cost inequality with the intrinsic distance. So, (1.4) was proved in [6] by using a known damped log-Sobolev inequality on the path space and finite-dimensional approximations. To ensure the smoothness of the approximating diffusions, one needs the boundedness of curvature. To get rid of this condition, a sequence of new metric approximating the original one were constructed in [6], which satisfy (1.1) and have bounded curvatures. In this way (1.4) was established without using curvature upper bounds. But to realize this approximation argument, the technical condition (1.3) with ∫_{0}^{∞} \frac{1}{ψ(s)}ds = ∞ was adopted. In this paper we adopt a different argument developed in [27] for diffusions on ℝ^{d} by using the martingale representation theorem and Girsanov transformations, so that this technical condition was avoided. Furthermore, we present a number of cost inequalities which are equivalent to the convexity of ∂M (if exists) and the curvature condition (1.1). When ∂M ≠ ∅, let N be the inward unit normal vector field of ∂M. Then the second fundamental form of ∂M is defined by

𝕀(U, V) = −⟨∇_{U}N, V ⟩, U, V ∈ T ∂M,

where T ∂M is the tangent space of ∂M. If 𝕀 ≥ 0, i.e. 𝕀(U, U) ≥ 0 for all U ∈ T ∂M, we call M (or ∂M) convex.

Theorem 1.1. Let P_{T} (o, ·) be the distribution of X_{T} with X_{0} = o, and let P_{T} be the corresponding semigroup. The following statements are equivalent to each other:

(1) ∂M is either convex or empty, and (1.1) holds.

(2) For any T > 0, μ ∈ 𝒫(M) and nonnegative F with Π_{μ}^{T} (F) = 1,

W_{2,ρ_{∞}}(F Π_{μ}^{T} , Π_{μ_{F}^{T}}^{T})^{2} ≤ \frac{2}{K} (e^{2KT} − 1)Π_{μ}^{T} (F log F)

holds, where μ_{F}^{T} ∈ 𝒫(M) is fixed by (1.2).

(3) (1.4) holds for any o ∈ M and T > 0.

(4) For any o ∈ M and T > 0,

W_{2,ρ}(P_{T} (o, ·), fP_{T} (o, ·))^{2} ≤ \frac{2}{K} (e^{2KT} − 1)P_{T} (f log f)(o),

f ≥ 0, P_{T} f(o) = 1.

(5) For any T > 0, μ, ν ∈ 𝒫(M), and p ≥ 1,

W_{p,ρ_{∞}}(Π_{μ}^{T} , Π_{ν}^{T}) ≤ e^{KT} W_{p,ρ}(μ, ν),

where W_{p,ρ} is the L^{p}-Wasserstein distance for probability measures on M induced by ρ.

(6) For any x, y ∈ M and T > 0,

W_{2,ρ}(P_{T} (x, ·), P_{T} (y, ·)) ≤ e^{KT} ρ(x, y).

(7) For any T > 0, μ ∈ 𝒫(M), and F ≥ 0 with Π_{μ}^{T} (F) = 1,

W_{2,ρ_{∞}}(F Π_{μ}^{T} , Π_{μ}^{T}) ≤ \{\frac{2}{K} (e^{2KT} − 1)Π_{μ}^{T} (F log F)\}^{1/2} + e^{KT} W_{2,ρ}(μ_{F}^{T} , μ).

(8) For any μ ∈ 𝒫(M) and C ≥ 0 such that

W_{2,ρ}(fμ, μ)^{2} ≤ Cμ(f log f), f ≥ 0, μ(f) = 1,

there holds

W_{2,ρ_{∞}}(F Π_{μ}^{T} , Π_{μ}^{T})^{2} ≤ (\sqrt{\frac{2}{K} (e^{2KT} − 1)} + e^{KT} \sqrt{C})^{2}Π_{μ}^{T} (F log F),

F ≥ 0, Π_{μ}^{T} (F) = 1.

When ∂M is empty or convex, there exist many equivalent semigroup inequalities for the curvature condition (1.1): see e.g. [3, 10] for equivalent statements on gradient estimates, log-Sobolev/Poicaré inequalities, and isoperimetric inequality; [19, 22, 23] for equivalent Harnack type inequalities; and [11] for equivalent inequalities on Wasserstein distances. The corresponding results have been partly extended in [23, 24] to the non-convex case. Theorem 1.1 provides seven equivalent inequalities for the convexity of ∂M (if exists) and the curvature condition (1.1), which are new even for manifolds without boundary. We would like to mention that the log-Sobolev inequality has been established in [26] on the path space over manifolds with convex boundary, and in a forthcoming paper we will extend the results on intrinsic ultracontractivity derived in [25] to non-compact manifolds with boundary. To prove Theorem 1.1, we shall use a formula of the second fundamental form established in [22] for compact manifolds with boundary. Since in this paper the manifold is allowed to be non-compact, we shall reprove this formula in Section 2 by using the reflecting diffusion process up to the exit time of a compact domain. This formula implies the equivalence of Theorem 1.1(1) and the semigroup log-Sobolev/Poincaré inequalities (see Theorem 2.4 below). In Section 3 we prove Theorem 1.1 by using results in Section 2, the martingale representation and Girsanov transformation for (reflecting) diffusions on (convex) manifolds. To establish transportation-cost inequalities on the path space for non-convex manifolds, we shall adopt a conformal change of metric ⟨·, ·⟩′ = f^{−2}⟨·, ·⟩ such that ∂M is convex under the new metric (see [21, Lemma 2.1]). Let Δ′ be the Laplacian induced by the new metric, we have (see [21, Lemma 2.2])

L = f^{−2}\{Δ′ + f^{2}Z + \frac{d − 2}{2}∇f^{2}\}.  (1.5)

According to this fact, we will modify our arguments in Section 4 to study the reflecting diffusion process generated by L := ψ^{2}(Δ + Z) for a smooth function ψ on a convex manifold, then extend Theorem 1.1 in Section 5 to the non-convex setting.

2 Formulae for the second fundamental form and applications

When M is compact, the following formula on ∂M has been found in [22]:

lim_{t→0} \frac{|∇f|^{2}}{\sqrt{t}} log \frac{|∇P_{t}f|}{(P_{t}|∇f|^{p})^{1/p}} = −\frac{2}{\sqrt{π}} 𝕀(∇f, ∇f), p ≥ 1,  (2.1)

where f is a smooth function satisfying the Neumann boundary condition. When M is non-compact, some technical problems appear in the original proof when e.g. a dominated convergence is used. To fix these problems, we shall stop the process in a compact domain, so that we shall first study the behavior of hitting times. Recall that the reflecting L-diffusion process can be constructed by solving the SDE

dX_{t} = \sqrt{2} Φ_{t} ∘ dB_{t} + Z(X_{t})dt + N(X_{t})dl_{t},  (2.2)

where Φ_{t} is the horizontal lift of X_{t} onto the frame bundle O(M), B_{t} is the d-dimensional Brownian motion. By the Itô formula, for any f ∈ C^{2}(M) we have

df(X_{t}) = \sqrt{2}⟨∇f(X_{t}), Φ_{t} ∘ dB_{t}⟩ + Lf(X_{t})dt + Nf(X_{t})dl_{t},  (2.3)

where Nf = ⟨N, ∇f⟩. For any R > 0, let

τ_{R} = inf{t ≥ 0 : ρ(X_{0}, X_{t}) ≥ R}.

Proposition 2.1. Let R > 0 and X_{0} = o ∈ M be fixed. Then there exist two constants c_{1}, c_{2} > 0 such that

ℙ(τ_{R} ≤ t) ≤ c_{1}e^{−c_{2}/t}, t > 0.

Proof. This result is well known on manifolds without boundary (cf. [2, Lemma 2.3]), and the proof works also when ∂M is convex. As in the present case the boundary is not necessarily convex, we shall follow [21] to make the boundary convex under a conformal change of metric. Since

B_{R} := {x ∈ M : ρ(o, x) ≤ R}

is compact, there exists a constant σ > 0 such that 𝕀 ≥ −σ holds on ∂M ∩ B_{R}. Let f ≥ 1 be smooth such that

N log f ≥ σ on ∂M ∩ B_{R}.  (2.4)

Such a function can be constructed by using the distance function ρ_{∂} to the boundary ∂M. Since ρ_{∂} is smooth in a neighborhood of the boundary, there exists a constant r_{0} > 0 such that ρ_{∂} is smooth on {x ∈ B_{2R} : ρ_{∂}(x) ≤ r_{0}}. Let h ∈ C^{∞}([0, ∞)) such that h′ ≥ 0, h(0) = 1, h′(0) = σ and h′(r) = 0 for r ≥ r_{0}. Then h ∘ ρ_{∂} is smooth on B_{2R} and N log h ∘ ρ_{∂}|_{∂M∩B_{2R}} = σ. Thus, it suffices to take smooth f ≥ 1 such that f = h ∘ ρ_{∂} on B_{R}. By [21, Lemma 2.1] and (2.4), ∂M is convex in B_{R} under the new metric

⟨·, ·⟩′ := f^{−2}⟨·, ·⟩,

where ⟨·, ·⟩ is the original metric. Let Δ′ be the Laplacian induced by the new metric. We have (see [21, Lemma 2.2])

L = f^{−2}(Δ′ + Z′)

for some C^{1}-vector field Z′. Let ρ̃_{o} be the Riemannian distance to o induced by the new metric. By the Laplacian comparison theorem,

Lρ̃_{o}^{2} ≤ c on B_{R}  (2.5)

holds for some constant c > 0 outside the cut-locus induced by ⟨·, ·⟩′. Since ∂M is convex on B_{R} and N is still the inward normal vector under the new metric, we have

Nρ̃_{o} ≤ 0 on ∂M ∩ B_{R}.

Therefore, by using Kendall’s Itô formula for the distance (cf. [9] for f = 1), (2.5) implies

dρ̃_{o}^{2}(X_{t}) ≤ 2\sqrt{2} f^{−2}(X_{t})ρ̃_{o}(X_{t})db_{t} + cdt, t ≤ τ_{R},

or equivalently,

t ↦ 2\sqrt{2} ∫_{0}^{t∧τ_{R}} f^{−2}(X_{s})ρ̃_{o}(X_{s})db_{s} + c(t ∧ τ_{R}) − ρ̃_{0}^{2}(X_{t∧τ_{R}})

is an increasing process, where b_{t} is some one-dimensional Brownian motion. Since f^{−2} ≤ 1, this implies that for any δ > 0, the process

Z_{s} := exp [\frac{δ}{t} ρ̃_{o}^{2}(X_{s}) − \frac{δcs}{t} − \frac{4δ^{2}}{t^{2}} ∫_{0}^{s} ρ̃_{o}^{2}(X_{u})du], s ≤ τ_{R}

is a super martingale. Therefore, letting C > 1 be a constant such that f ≤ C on B_{R} and thus, ρ_{o} ≥ ρ̃_{o} ≥ C^{−1}ρ_{o} holds on B_{R}, we obtain

ℙ(τ_{R} ≤ t) = ℙ( max_{s∈[0,t]} ρ_{o}(X_{s∧τ_{R}}) ≥ R) ≤ ℙ(R ≥ max_{s∈[0,t]} ρ̃_{o}(X_{s∧τ_{R}}) ≥ \frac{R}{C} )

≤ ℙ( max_{s∈[0,t]} Z_{s∧τ_{R}} ≥ exp [\frac{δR^{2}}{tC^{2}} − δc − \frac{4δ^{2}R^{2}}{t} ])

≤ exp [cδ − \frac{R^{2}}{tC^{2}} (δ − 4C^{2}δ^{2})], δ > 0.

The proof is then completed by taking e.g. δ = 1/(8C^{2}).

Proposition 2.2. Let X_{0} = o ∈ ∂M. Then for any R > 0,

lim sup_{t→0} \frac{1}{t}|𝔼l_{t∧τ_{R}} − 2\sqrt{t/π}| < ∞.

Proof. Repeating the proof of [22, Lemma 2.2] by using t ∧ τ_{R} in place of t, we obtain

𝔼l_{t∧τ_{R}}^{2} ≤ ct, t ∈ [0, 1]  (2.6)

for some constant c > 0. Let r_{0} > 0 be such that ρ_{∂} is smooth on {ρ_{∂} ≤ r_{0}}∩B_{R}. Let

τ = inf{t ≥ 0 : ρ_{∂}(X_{t}) ≥ r_{0}}.

By the Itô formula we have

dρ_{∂}(X_{t}) = \sqrt{2} db_{t} + Lρ_{∂}(X_{t})dt + dl_{t}, t ≤ τ ∧ τ_{R},  (2.7)

where, as before, b_{t} is some one-dimensional Brownian motion. By the proof of [22, Theorem 2.1] using τ ∧ τ_{R} in place of τ, we have, instead of (2.4) in [22],

𝔼(ρ_{∂}(X_{t∧τ∧τ_{R}}) − \sqrt{2} |b̃_{t∧τ∧τ_{R}}|)^{2} ≤ c_{1}t^{2}, t ∈ [0, 1]  (2.8)

for some constant c_{1} > 0, where b̃_{t} is some one-dimensional Brownian motion. Due to (2.7),

|𝔼l_{t∧τ∧τ_{R}} − 𝔼ρ_{∂}(X_{t∧τ∧τ_{R}})| ≤ c_{2}t

holds for some constant c_{2} > 0. Combining this with (2.8) we arrive at

|𝔼l_{t∧τ∧τ_{R}} − \sqrt{2} 𝔼|b̃_{t∧τ∧τ_{R}}|| ≤ c_{3}t, t ∈ [0, 1] for some constant c_{3} > 0. Since 𝔼|b̃_{t}| = \sqrt{2t/π} and 𝔼|b̃_{t}|^{2} = t, this and (2.6) imply

\begin{matrix}|𝔼l_{t∧τ_{R}} − \frac{2\sqrt{t}}{\sqrt{π}}| = |𝔼l_{t∧τ_{R}} − \sqrt{2} 𝔼|b̃_{t}|| \\ ≤ c_{3}t + 𝔼1_{\{t≥τ∧τ_{R}\}}(l_{t∧τ_{R}} + \sqrt{2} |b̃_{t}|) \\ ≤ c_{3}t + c_{4}\sqrt{tℙ(t ≥ τ ∧ τ_{R})}, t ∈ [0, 1]. \end{matrix}  (2.9)

Moreover, noting that

ℙ(τ ∧ τ_{R} ≤ t, τ_{R} > τ) ≤ ℙ( max_{s∈[0,t]} ρ_{∂}(X_{s∧τ∧τ_{R}}) ≥ r_{0}),

by using τ ∧ τ_{R} to replace τ in the proof of [22, Proposition A.2], we conclude that

ℙ(τ ∧ τ_{R} ≤ t, τ_{R} > τ) ≤ c_{5} exp[−r_{0}^{2}/(16t)], t > 0

holds for some constant c_{5} > 0. Combining this with Proposition 2.1, we obtain

ℙ(t ≥ τ ∧ τ_{R}) ≤ c_{6}e^{−c_{7}/t}, t > 0

for some constants c_{6}, c_{7} > 0. Therefore, the proof is completed by (2.9).

Theorem 2.3. Let f ∈ C^{∞}(M) with Nf|_{∂M} = 0.

(1) For any p ≥ 1 and R > 0,

lim_{t→0} \frac{|∇f|^{2}}{\sqrt{t}} log \frac{(𝔼|∇f|^{p}(X_{t∧τ_{R}})|)^{1/p}}{|∇f|} = \frac{2}{\sqrt{π}} 𝕀(∇f, ∇f)  (2.10)

holds at points on ∂M such that |∇f| > 0.

(2) Assume that for any g ∈ C_{0}^{1}(M) the function |∇P_{·}g| is bounded on [0, 1] × M. If moreover f has a compact support, then (2.1) holds points on ∂M such that |∇f| > 0.

Proof. (2.10) follows immediately from the proof of [22, Theorem 1.2] by using Proposition 2.2 in place of [22, Theorem 2.1], and using t ∧ τ_{R} in place of t. Next, let f ∈ C_{0}^{∞}(M). By the assumption of (2) and that Lf ∈ C_{0}^{1}(M), |∇P_{·}Lf| is bounded on [0, 1] × M. So, the proof of [22, (3.1)] implies that

lim_{t→0} \frac{|∇f|^{2}}{\sqrt{t}} log \frac{|∇P_{t}f|}{(P_{t}|∇f|^{p})^{1/p}} = − lim_{t→0} \frac{|∇f|^{2}}{\sqrt{t}} log \frac{(P_{t}|∇f|^{p})^{1/p}}{|∇f|} .  (2.11)

Since by Proposition 2.1, there exist two constant c_{1}, c_{2} > 0 such that

|P_{t}|∇f|^{p} − 𝔼|∇f|^{p}(X_{t∧τ_{R}})| ≤ ∥∇f∥_{∞}^{p}ℙ(t > τ_{R}) ≤ c_{1}e^{−c_{2}/t}, t > 0,

we conclude that (2.1) follows from (2.11) and (2.10).

As an application of (2.10), the following result provides equivalent semigroup log-Sobolev/Poincaré inequalities for Theorem 1.1(1).

Theorem 2.4. Each of the following statements is equivalent to Theorem 1.1(1):

(9) For any T > 0 and f ∈ C_{b}(M),

P_{T} f^{2} log f^{2} ≤ (P_{T} f^{2}) log P_{T} f^{2} + \frac{e^{2KT} − 1}{2K}P_{T} |∇f|^{2}.

(10) For any T > 0 and f ∈ C_{b}(M),

P_{T} f^{2} ≤ (P_{T} f)^{2} + \frac{e^{2KT} − 1}{K}P_{T} |∇f|^{2}.

Proof. According to e.g. [16, Lemma 3.1], which holds also for the nonsymmetric case, Theorem 1.1(1) implies the semigroup log-Sobolev inequality (9). It is well known that the log-Sobolev inequality implies the Poincaré inequality. So, (10) follows from (9). Hence, it remains to show that (10) implies Theorem 1.1(1). Below we shall prove the convexity of ∂M and the curvature condition (1.1) respectively.

(a) Let ∂M ≠ ∅. For any o ∈ ∂M and non-trivial U ∈ T_{o}∂M, we aim to show that 𝕀(U, U) ≥ 0. Let f ∈ C_{b}^{∞}(M) such that Nf|_{∂M} = 0 and ∇f(o) = U. Let X_{0} = o and

τ_{1} = inf{t ≥ 0 : ρ(o, X_{t}) ≥ 1}.

Since f and f^{2} satisfies the Neumann boundary condition, we have

\begin{matrix}𝔼f(X_{t∧τ_{1}}) = f(o) + 𝔼 ∫_{0}^{t∧τ_{1}} Lf(X_{s})ds, \\ 𝔼f^{2}(X_{t∧τ_{1}}) = f^{2}(o) + 2𝔼 ∫_{0}^{t∧τ_{1}}(fLf)(X_{s})ds + 2𝔼 ∫_{0}^{t∧τ_{1}} |∇f|^{2}(X_{s})ds. \end{matrix}

So,

\begin{matrix}𝔼f^{2}(X_{t∧τ_{1}}) − \{𝔼f(X_{t∧τ_{1}})\}^{2} = 2𝔼 ∫_{0}^{t∧τ_{1}}\{f(X_{s}) − f(X_{0})\}Lf(X_{s})ds \\ − (𝔼 ∫_{0}^{t∧τ_{1}} Lf(X_{s})ds)^{2} + 2𝔼 ∫_{0}^{t∧τ_{1}} |∇f|^{2}(X_{s})ds. \end{matrix} (2.12)

Since Lf is bounded on B_{1} := {x : ρ(o, x) ≤ 1}, we have

(𝔼 ∫_{0}^{t∧τ_{1}} Lf(X_{s})ds)^{2} ≤ ct^{2}  (2.13)

for some c > 0. Moreover, due to Proposition 2.1,

ℙ(τ_{1} ≤ t) ≤ c_{1}e^{−c_{2}/t}, t > 0  (2.14)

holds for some constants c_{1}, c_{2} > 0. Thus,

\begin{matrix}|P_{t}f^{2}(o) − (P_{t}f)^{2}(o) − (𝔼f^{2}(X_{t∧τ_{1}}) − \{𝔼f(X_{t∧τ_{1}})\}^{2})| = o(t^{2}), \\ 𝔼 ∫_{0}^{t∧τ_{1}} |∇f|^{2}(X_{s})ds = t|∇f(o)|^{2} + ∫_{0}^{t}𝔼\{|∇f|^{2}(X_{s∧τ_{1}}) − |∇f(o)|^{2}\}ds + o(t^{2}), \end{matrix} ; (2.15)

where and in what follows, o(s) stands for a function of s > 0 such that lim_{s→0} o(s)/s = 0. Similarly, applying the Itô formula to {f(X_{s}) − f(o)}Lf(X_{s}), we obtain (note that Nf|_{∂M} = 0)

𝔼 ∫_{0}^{t∧τ_{1}}{f(X_{s}) − f(o)}Lf(X_{s})ds
= o(t^{2}) + ∫_{0}^{t} 𝔼[(f(X_{s∧τ_{1}}) − f(o))Lf(X_{s∧τ_{1}})]ds
= o(t^{2}) + 𝔼 ∫_{0}^{t} ds ∫_{0}^{s∧τ_{1}} L{(f − f(o))Lf}(X_{r})dr
+ 𝔼 ∫_{0}^{t} ds ∫_{0}^{s∧τ_{1}}{(f − f(o))NLf}(X_{r})dl_{r}.

(2.16)

Noting that

f(X_{r}) − f(o) = \sqrt{2} ∫_{0}^{r}⟨∇f(X_{u}), Φ_{u} ∘ dB_{u}⟩ + ∫_{0}^{r} Lf(X_{u})du, u ≤ τ_{1},

and that

𝔼 sup_{r∈[0,t]} ( ∫_{0}^{r}⟨∇f(X_{u}), Φ_{u} ∘ dB_{u}⟩)^{2} ≤ c_{2}t, t ∈ [0, 1]

holds for some constant c_{2} > 0, we obtain from (2.16) and (2.6) that

|𝔼 ∫_{0}^{t∧τ_{1}}{f(X_{s}) − f(o)}Lf(X_{s})ds| ≤ c_{3}t^{2}, t ∈ [0, 1]  (2.17)

holds for some constant c_{3} > 0. Finally, by Theorem 2.3(1), we have

𝔼|∇f|^{2}(X_{t∧τ_{1}}) = |∇f|^{2}(o) + \frac{4\sqrt{t}}{\sqrt{π}}𝕀(∇f, ∇f)(o) + o(t^{1/2})  (2.18)

for small t > 0. Combining this with (2.12), (2.13), (2.15) and (2.17), and noting that U = ∇f(o), we conclude that

P_{t}f^{2}(o) − (P_{t}f)^{2}(o) = 2t|∇f(o)|^{2} + \frac{16t^{3/2}}{3\sqrt{π}} 𝕀(U, U) + o(t^{3/2}).  (2.19)

Finally, (2.18) and (2.14) imply that

\frac{e^{2Kt} − 1}{K}P_{t}|∇f|^{2}(o) = 2t|∇f(o)|^{2} + \frac{8t^{3/2}}{\sqrt{π}} 𝕀(U, U) + o(t^{3/2}).

Since \frac{16}{3} < 8, combining this with (10) and (2.19) we conclude that 𝕀(U, U) ≥ 0.

(b) Let X_{0} = o ∈ M \∂M, we aim to show that Ric−∇Z ≥ −K holds on T_{o}M. Let R > 0 such that B_{R} ∩ ∂M = ∅. Since l_{t} increases only when X_{t} ∈ ∂M, l_{t} = 0 for t ≤ τ_{R}. Hence, due to Proposition 2.1, for any f ∈ C_{b}^{∞}(M),

\begin{matrix}P_{t}f^{2}(o) − (P_{t}f)^{2}(o) = o(t^{2}) + 𝔼f^{2}(X_{t∧τ_{R}}) − (𝔼f(X_{t∧τ_{R}}))^{2} \\ = o(t^{2}) + ∫_{0}^{t}\{𝔼Lf^{2}(X_{s∧τ_{R}}) − 2f(o)𝔼Lf(X_{s∧τ_{R}})\}ds − ( ∫_{0}^{t}𝔼Lf(X_{s∧τ_{R}})ds)^{2}. \end{matrix} ; (2.20)

By the continuity of s ↦ Lf(X_{s∧τ_{R}}), we have

( ∫_{0}^{t} 𝔼Lf(X_{s∧τ_{R}})ds)^{2} = (Lf)^{2}(o)t^{2} + o(t^{2}).  (2.21)

Similarly, it is easy to see that

𝔼Lf^{2}(X_{s∧τ_{R}}) − 2f(o)𝔼Lf(X_{s∧τ_{R}})
= Lf^{2}(o) − 2f(o)Lf(o) + s{LLf^{2} − 2fLLf}(o) + o(s)
= 2|∇f|^{2}(o) + 2s{L|∇f|^{2}(o) + (Lf)^{2}(o) + 2⟨∇f, ∇Lf⟩(o)} + o(s).

Combining this with (2.20) and (2.21) we obtain

P_{t}f^{2}(o)−(P_{t}f)^{2}(o) = 2t|∇f|^{2}(o)+t^{2}(L|∇f|^{2}+2⟨∇f, ∇Lf⟩}(o)+o(t^{2}). (2.22)

Finally, by Proposition 2.1 and noting that l_{s} = 0 for s ≤ τ_{R}, we have

P_{t}|∇f|^{2}(o) = o(t^{2}) + 𝔼|∇f|^{2}(X_{t∧τ_{R}}) = |∇f|^{2}(o) + tL|∇f|^{2}(o) + o(t).

Combining this with (10) and (2.22), we conclude that

\frac{1}{2}L|∇f|^{2}(o) − ⟨∇f, ∇Lf⟩(o) ≥ −K|∇f|(o), f ∈ C_{b}^{∞}(M).

This completes the proof by the Bochner-Weitzenböck formula.

3 Proof of Theorem 1.1

By taking μ = δ_{o}, we have μ_{F}^{T} = Π_{o}^{T} (F)δ_{o} = δ_{o}. So, (3) follows from each of (2), (7) and (8). Next, (4) follows from (3) by taking F (X_{[0,T]}) = f(X_{T}), and (5) implies (6) by taking p = 2 and μ = δ_{x}, ν = δ_{y}. Moreover, it is clear that (8) follows from (7) while (7) is implied by (2) and (5). So, it suffices to prove that (1) ⇒ (3) ⇒ (2), (4) ⇒ (1) ⇒ (6) ⇒ (5) and (6) ⇒ (1), where “ ⇒” stands for “implies”.

(a) (1) ⇒ (3). We shall only consider the case where ∂M is non-empty and convex. For the case without boundary, the following argument works well by taking l_{t} = 0 and N = 0. The idea of the proof comes from [27], where elliptic diffusions on ℝ^{d} were concerned. Let B_{t} be the d-dimensional Brownian motion on the naturally filtered probability space (Ω, ℱ_{t}, ℙ). Let {X_{t} : t ≥ 0} solve (2.2) with X_{0} = o. Next, let F be a positive bounded measurable function on M^{T} such that inf F > 0 and Π_{o}^{T} (F) = 1. Then

m_{t} := 𝔼_{ℙ}(F (X_{[0,T]})|ℱ_{t}) and L_{t} := ∫_{0}^{t} \frac{dm_{s}}{m_{s}} , t ∈ [0, T]

are square-integrable ℱ_{t}-martingales under ℙ, where 𝔼_{ℙ} is the expectation taken for the probability measure ℙ. Obviously, we have

m_{t} = e^{L_{t}− \frac{1}{2} ⟨L⟩_{t}}, t ∈ [0, T].  (3.1)

Since ℱ_{t} is the natural filtration of B_{t}, by the martingale representation theorem (cf. [8, Theorem 6.6]), there exists a unique ℱ_{t}-predictable process β_{t} on ℝ^{d} such that

L_{t} = ∫_{0}^{t}⟨β_{s}, dB_{s}⟩, t ∈ [0, T].  (3.2)

Let dℚ = F (X_{[0,T]})dℙ. Since 𝔼_{ℙ}F (X_{[0,T]}) = Π_{μ}^{T} (F) = 1, ℚ is a probability measure on Ω. Due to (3.1) and (3.2) we have

F (X_{[0,T]}) = m_{T} = e^{∫_{0}^{T} ⟨β_{s},dB_{s}⟩− \frac{1}{2} ∫_{0}^{T} ∥β_{s}∥^{2}ds}.

Moreover, by the Girsanov theorem,

B̃_{t} := B_{t} − ∫_{0}^{t} β_{s}ds, t ∈ [0, T]  (3.3)

is a d-dimensional Brownian motion under the probability measure ℚ. Let Y_{t} solve the SDE

dY_{t} = \sqrt{2} P_{X_{t},Y_{t}}Φ_{t} ∘ dB̃_{t} + Z(Y_{t})dt + N(Y_{t})dl̃_{t}, Y_{0} = o,  (3.4)

where P_{X_{t},Y_{t}} is the parallel displacement along the minimal geodesic from X_{t} to Y_{t} and l̃_{t} is the local time of Y_{t} on ∂M. As explained in e.g. [1, Section 3], we may assume that the minimal geodesic is unique so that P_{x,y} is smooth in x, y ∈ M. Since, under ℚ, B̃_{t} is a d-dimensional Brownian motion, the distribution of Y_{[0,T]} is Π_{o}^{T}. On the other hand, by (2.2) and (3.3), we have

dX_{t} = \sqrt{2} Φ_{t} ∘ dB̃_{t} + Z(X_{t}) + \sqrt{2} Φ_{t}β_{t}dt + N(X_{t})dl_{t}.  (3.5)

Since for any bounded measurable function G on M^{T}

𝔼_{ℚ}G(X_{[0,T]}) = 𝔼_{ℙ}(F G)(X_{[0,T]}) = Π_{o}^{T} (F G),

we conclude that under ℚ the distribution of X_{[0,T]} coincides with F Π_{μ}^{T}. Therefore,

W_{2,ρ_{∞}}(F Π_{o}^{T} , Π_{o}^{T})^{2} ≤ 𝔼_{ℚ}ρ_{∞}(X_{[0,T]}, Y_{[0,T]})^{2} = 𝔼_{ℚ} max_{t∈[0,T]} ρ(X_{t}, Y_{t})^{2}. (3.6)

By the convexity of ∂M we have

⟨N(x), ∇ρ(y, ·)(x)⟩ = ⟨N(x), ∇ρ(·, y)(x)⟩ ≤ 0, x ∈ ∂M.

Combining this with the Itô formula for (X_{t}, Y_{t}) given by (3.4) and (3.5), we obtain from (1.1) that

\begin{aligned}dρ(X_{t}, Y_{t}) & ≤ Kρ(X_{t}, Y_{t})dt + \sqrt{2} ⟨Φ_{t}β_{t}, ∇ρ(·, Y_{t})(X_{t})⟩dt \\ & ≤ (Kρ(X_{t}, Y_{t}) + \sqrt{2} ∥β_{t}∥)dt, \end{aligned}

see e.g. [15, Lemmas 2.1 and 2.2]. Since we are using the coupling by parallel displacement instead of the mirror reflection, the martingale part here disappears (cf. Theorem 2 and (2.5) in [9]). Since X_{0} = Y_{0}, this implies

ρ(X_{t}, Y_{t})^{2} ≤ e^{2Kt}(\sqrt{2} ∫_{0}^{t} e^{−Ks}∥β_{s}∥ ds)^{2} ≤ \frac{e^{2Kt} − 1}{K} ∫_{0}^{t} ∥β_{s}∥^{2}ds, t ∈ [0, T].

Therefore,

𝔼_{ℚ} max_{t∈[0,T]} ρ(X_{t}, Y_{t})^{2} ≤ \frac{e^{2KT} − 1}{K} ∫_{0}^{T} 𝔼_{ℚ}∥β_{s}∥^{2}ds.  (3.7)

It is clear that

\begin{matrix}𝔼_{ℚ}∥β_{s}∥^{2} = 𝔼_{ℙ}(m_{T} ∥β_{s}∥^{2}) \\ = 𝔼_{ℙ}(∥β_{s}∥^{2}𝔼_{ℙ}(m_{T} |ℱ_{s})) = 𝔼_{ℙ}(m_{s}∥β_{s}∥^{2}), s ∈ [0, T]. \end{matrix}  (3.8)

Finally, since (3.1) and (3.2) yield

d⟨m⟩_{t} = m_{t}^{2}d⟨L⟩_{t} = m_{t}^{2}∥β_{t}∥^{2}dt,

we have

dm_{t} log m_{t} = (1 + log m_{t})dm_{t} + \frac{d⟨m⟩_{t}}{2m_{t}}

= (1 + log m_{t})dm_{t} + \frac{m_{t}}{2} ∥β_{t}∥^{2}dt.

As m_{t} is a ℙ-martingale, combining this with (3.8) we obtain

∫_{0}^{T} 𝔼_{ℚ}∥β_{s}∥^{2}ds = 2𝔼_{ℙ}F (X_{[0,T]}) log F (X_{[0,T]}).  (3.9)

Therefore, (1.4) follows from (3.6), (3.7) and (3.9).

(b) (3) ⇒ (2). By (3), for each x ∈ M, there exists

π_{x} ∈ 𝒞 (\frac{F}{Π_{x}^{T} (F)}Π_{x}^{T} , Π_{x}^{T} )

such that

∫_{M^{T} ×M^{T}} ρ_{∞}(γ, η)^{2}π_{x}(dγ, dη) ≤ \frac{2}{K} (e^{2KT} −1)Π_{x}^{T} (\frac{F}{Π_{x}^{T} (F)} log \frac{F}{Π_{x}^{T} (F)}). (3.10)

If x ↦ π_{x}(G) is measurable for bounded continuous functions G on M^{T} × M^{T}, then

π := ∫_{M} π_{x}μ_{F}^{T} (dx) ∈ 𝒞 (F Π_{μ}^{T} , Π_{μ_{F}^{T}}^{T})

is well defined and by (3.10)

∫_{M^{T} ×M^{T}} ρ_{∞}^{2}dπ ≤ \frac{2}{K} (e^{2KT} − 1) ∫_{M} Π_{x}^{T} (F log \frac{F}{Π_{x}^{T} (F)})μ(dx)

≤ \frac{2}{K} (e^{2KT} − 1)Π_{μ}^{T} (F log F).

This implies the inequality in (2). To confirm the measurability of x ↦ π_{x}, we first consider discrete μ, i.e. μ = ∑_{n=1}^{∞} ε_{n}δ_{x_{n}} for some {x_{n}} ⊂ M and ε_{n} ≥ 0 with ∑_{n=1}^{∞} ε_{n} = 1. In this case

π_{x} = ∑_{n=1}^{∞} 1_{{x=x_{n}}}π_{x_{n}}, μ-a.e.

which is measurable in x and π = ∑_{n=1}^{∞} μ_{F}^{T} ({x_{n}})π_{x_{n}}. Hence, the inequality in (2) holds. Then, for general μ, the desired inequality can be derived by approximating μ with discrete distributions in a standard way, see (b) in the proof of [6, Theorem 4.1].

(c) (4) ⇒ (1). According to [12, Section 7] (see also [4, Section 4.1]), by first applying the transportation-cost inequality in (3) to 1 − ε + εf in place of f, then letting ε → 0, we obtain the Poincaré inequality

P_{T} f^{2} ≤ \frac{e^{2KT} − 1}{K}P_{T} |∇f|^{2} + (P_{T} f)^{2}, f ∈ C_{b}^{1}(M), T > 0.  (3.11)

Thus, the proof is finished by Theorem 2.4.

(d) (1) ⇒ (6). Let X_{t} solve (2.2) with X_{0} = x and Y_{t} solve

dY_{t} = \sqrt{2} P_{X_{t},Y_{t}}Φ_{t} ∘ dB_{t} + Z(Y_{t})dt − N(Y_{t})dl̃_{t}, Y_{0} = y,  (3.12)

where l̃_{t} is the local time of Y_{t} on ∂M. Since ∂M is convex and (1.1) holds, as explained in (a), we have

dρ(X_{t}, Y_{t}) ≤ Kρ(X_{t}, Y_{t})dt.

Thus, ρ_{∞}(X_{·}, Y_{·}) ≤ e^{KT} ρ(x, y). This implies (6).

(e) (6) ⇒ (5). By (6), for any x, y ∈ M, there exists π_{x,y} ∈ 𝒞 (Π_{x}^{T} , Π_{y}^{T}) such that

∫_{M^{T} ×M^{T}} ρ_{∞}^{p}dπ_{x,y} ≤ e^{KT} ρ(x, y)^{p}.

As explained in (b), we assume that μ and ν are discrete, so that for any π^{0} ∈ (μ, ν), π_{x,y} has a π^{0}-version measurable in (x, y). Thus,

π := ∫_{M×M} π_{x,y}π^{0}(dx, dy) ∈ 𝒞 (Π_{μ}^{T} , Π_{ν}^{T})

satisfies

∫_{M^{T} ×M^{T}} ρ_{∞}^{p}dπ ≤ e^{KT} ∫_{M×M} ρ(x, y)^{p}π^{0}(dx, dy).

This implies the desired inequality in (5).

(f) (6) ⇒ (1). Let T > 0 be fixed. For any x, y ∈ M, let π_{x,y} ∈ 𝒞 (P_{T} (x, ·), P_{T} (y, ·)) be the optimal coupling for W_{2,ρ}, i.e.

W_{2,ρ}(P_{T} (x, ·), P_{T} (y, ·))^{2} = ∫_{M×M} ρ^{2}dπ_{x,y}.  (3.13)

Then for any f ∈ C_{b}^{2}(M), (6) implies

\begin{matrix}\frac{|P_{T} f(x) − P_{T} f(y)|}{ρ(x, y)} ≤ ∫_{M×M} \frac{|f(z_{1}) − f(z_{2})|}{ρ(z_{1}, z_{2})} · \frac{ρ(z_{1}, z_{2})}{ρ(x, y)}π_{x,y}(dz_{1}, dz_{2}) \\ ≤ \frac{W_{2,ρ}(P_{T} (x, ·), P_{T} (y, ·))}{ρ(x, y)}\{ ∫_{M×M} \frac{(f(z_{1}) − f(z_{2}))^{2}}{ρ(z_{1}, z_{2})^{2}} π_{x,y}(dz_{1}, dz_{2})\}^{1/2} \\ ≤ e^{KT} \{ ∫_{M×M} \frac{(f(z_{1}) − f(z_{2}))^{2}}{ρ(z_{1}, z_{2})^{2}} π_{x,y}(dz_{1}, dz_{2})\}^{1/2}. \end{matrix} ; (3.14)

Noting that f ∈ C_{b}^{2}(M) implies

|f(z_{1}) − f(z_{2})|^{2} ≤ ρ(z_{1}, z_{2})^{2}|∇f|^{2}(z_{1}) + cρ(z_{1}, z_{2})^{3}

for some constant c > 0, by (6) and (3.13) we obtain

∫_{M×M} \frac{(f(z_{1}) − f(z_{2}))^{2}}{ρ(z_{1}, z_{2})^{2}} π_{x,y}(dz_{1}, dz_{2}) ≤ P_{T} |∇f|^{2}(x) + ce^{KT} ρ(x, y).

Therefore, letting y → x in (3.14) we arrive at

|∇P_{T} f(x)| ≤ e^{KT} (P_{T} |∇f|^{2}(x))^{1/2}.

By a standard argument of Bakry and Emery, this implies the Poincaré inequality (3.11). Thus, (1) holds according to Theorem 2.4.

4 The case with a diffusion coefficient

Let ψ > 0 be a smooth function on M, and let Π_{μ,ψ}^{T} be the distribution of the (reflecting if ∂M ≠ ∅) diffusion process generated by L_{ψ} := ψ^{2}(Δ + Z) on time interval [0, T] with initial distribution μ, and let Π_{x,ψ}^{T} = Π_{δ_{x},ψ}^{T} for x ∈ M. Moreover, for F ≥ 0 with Π_{μ,ψ}^{T}(F) = 0, let

μ_{F,ψ}^{T}(dx) = Π_{x,ψ}^{T}(F)μ(dx).

Theorem 4.1. Assume that ∂M is either empty or convex and let (1.1) hold. Let ψ ∈ C_{b}^{∞}(M) be strictly positive. Let

K_{ψ} = K^{+}∥ψ∥_{∞}^{2} + 2∥Z∥_{∞}∥∇ψ∥_{∞}∥ψ∥_{∞} + (d − 1)∥∇ψ∥_{∞}^{2}.

Then

W_{2,ρ_{∞}}(F Π_{μ,ψ}^{T}, Π_{μ_{F,ψ}^{T},ψ}^{T})^{2} ≤ 2C(T, ψ)Π_{μ,ψ}^{T}(F log F),
μ ∈ 𝒫(M), F ≥ 0, Π_{μ,ψ}^{T}(F) = 1

holds for

C(T, ψ) := inf_{R>0} \{(1 + R^{−1})∥ψ∥_{∞}^{2} \frac{e^{2K_{ψ}T} − 1}{K_{ψ}} exp [2(1 + R)∥∇ψ∥_{∞}^{2} \frac{e^{2K_{ψ}T} − 1}{K_{ψ}}]\}.

Proof. As explained in (a) of the proof of Theorem 1.1, we shall only consider the case that ∂M is non-empty and convex. According to the proof of “(3) ⇒ (2)”, it suffices to prove for μ = δ_{o}, o ∈ M. In this case the desired inequality reduces to

W_{2,ρ_{∞}}(F Π_{o,ψ}^{T}, Π_{o,ψ}^{T}) ≤ 2C(T, ψ)Π_{o,ψ}^{T}(F log F), F ≥ 0, Π_{o,ψ}^{T}(F) = 1. (4.1)

Since the diffusion coefficient is non-constant, it is convenient to adopt the Itô differential d_{I} for the Girsanov transformation. So, the reflecting diffusion process generated by L_{ψ} := ψ^{2}(Δ + Z) can be constructed by solving the Itô SDE

d_{I}X_{t} = \sqrt{2} ψ(X_{t})Φ_{t}dB_{t} + ψ^{2}(X_{t})Z(X_{t})dt + N(X_{t})dl_{t},  (4.2)

where X_{0} = o and B_{t} is the d-dimensional Brownian motion with natural filtration ℱ_{t}. Let β_{t}, ℚ and B̃_{t} be fixed in the proof of Theorem 1.1. Then

d_{I}X_{t} = \sqrt{2} ψ(X_{t})Φ_{t}dB̃_{t} + \{ψ^{2}(X_{t})Z(X_{t}) + \sqrt{2} ψ(X_{t})Φ_{t}β_{t}\}dt + N(X_{t})dl_{t}.

(4.3)

Let Y_{t} solve

d_{I}Y_{t} = \sqrt{2} ψ(Y_{t})P_{X_{t},Y_{t}}Φ_{t}dB̃_{t} + ψ^{2}(Y_{t})Z(Y_{t})dt + N(Y_{t})dl̃_{t}, Y_{0} = o, (4.4)

where l̃_{t} is the local time of Y_{t} on ∂M. As in (a) of the proof of Theorem 1.1, under ℚ, the distributions of Y_{[0,T]} and X_{[0,T]} are Π_{o,ψ}^{T} and F Π_{o,ψ}^{T} respectively. So,

W_{2,ρ_{∞}}(F Π_{o,ψ}^{T}, Π_{o,ψ}^{T})^{2} ≤ 𝔼_{ℚ} max_{t∈[0,T]} ρ(X_{t}, Y_{t})^{2}.  (4.5)

Noting that due to the convexity of ∂M

⟨N(x), ∇ρ(y, ·)(x)⟩ = ⟨N(x), ∇ρ(·, y)(x)⟩ ≤ 0, x ∈ ∂M,

by (4.3), (4.4) and the Itô formula, we obtain

\begin{matrix}dρ( & X_{t}, Y_{t}) ≤ \sqrt{2} \{ψ(X_{t})⟨∇ρ(·, Y_{t})(X_{t}), Φ_{t}dB̃_{t}⟩ \\ + ψ(Y_{t})⟨∇ρ(X_{t}, ·)(Y_{t}), P_{X_{t},Y_{t}}Φ_{t}dB̃_{t}⟩\} \\ + \{ ∑_{i=1}^{d−1} U_{i}^{2}ρ(X_{t}, Y_{t}) + ⟨ψ(X_{t})^{2}Z(X_{t}) + \sqrt{2} ψ(X_{t})Φ_{t}β_{t}, ∇ρ(·, Y_{t})(X_{t})⟩ \\ + ψ(Y_{t})^{2}⟨Z(Y_{t}), ∇ρ(X_{t}, ·)(Y_{t})⟩\}dt, \end{matrix} ; (4.6)

where {U_{i}}_{i=1}^{d−1} are vector fields on M × M such that ∇U_{i}(X_{t}, Y_{t}) = 0 and

U_{i}(X_{t}, Y_{t}) = ψ(X_{t})V_{i} + ψ(Y_{t})P_{X_{t},Y_{t}}V_{i}, 1 ≤ i ≤ d − 1

for {V_{i}}_{i=1}^{d} an OBN of T_{X_{t}}M with V_{d} = ∇ρ(·, Y_{t})(X_{t}). In order to calculate U_{i}^{2}ρ(X_{t}, Y_{t}), we adopt the second variational formula for the distance. Let ρ_{t} = ρ(X_{t}, Y_{t}) and let {J_{i}}_{i=1}^{d−1} be Jacobi fields along the minimal geodesic γ : [0, ρ_{t}] → M from X_{t} to Y_{t} such that J_{i}(0) = ψ(X_{t})V_{i} and J_{i}(ρ_{t}) = ψ(Y_{t})P_{X_{t},Y_{t}}V_{i}, 1 ≤ i ≤ d − 1. Note that the existence of γ is ensured by the convexity of ∂M. Then, by the second variational formula and noting that ∇U_{i}(X_{t}, Y_{t}) = 0, we have

I := ∑_{i=1}^{d−1} U_{i}^{2}ρ(X_{t}, Y_{t}) = ∑_{i=1}^{d−1} ∫_{0}^{ρ_{t}} {|∇γ̇J_{i}|^{2} − ⟨ℛ(γ̇, J_{i})J_{i}, γ̇⟩}(s)ds,  (4.7)

where ℛ is the curvature tensor. Let

J̃_{i}(s) = (\frac{s}{ρ_{t}} ψ(Y_{t}) + \frac{ρ_{t} − s}{ρ_{t}}ψ(X_{t}))P_{γ(0),γ(s)}V_{i}, 1 ≤ i ≤ d − 1.

We have J̃_{i}(0) = J_{i}(0) and J̃_{i}(ρ_{t}) = J_{i}(ρ_{t}), 1 ≤ i ≤ i − 1. By the index lemma,

\begin{aligned}I & ≤ ∑_{i=1}^{d−1} ∫_{0}^{ρ_{t}} \{|∇γ̇ J̃_{i}|^{2} − ⟨ℛ(γ̇, J̃_{i})J̃_{i}, γ̇⟩\}(s)ds \\ & ≤ (d − 1)∥∇ψ∥_{∞}^{2}ρ_{t} − \frac{1}{ρ_{t}^{2}} ∫_{0}^{ρ_{t}} \{sψ(Y_{t}) + (ρ_{t} − s)ψ(X_{t})\}^{2}Ric(γ̇(s), γ̇(s))ds. \end{aligned} ; (4.8)

Moreover,

\begin{matrix}ψ(X_{t})^{2}⟨Z(X_{t}), ∇ρ(·, Y_{t})(X_{t})⟩ + ψ(Y_{t})^{2}⟨Z(Y_{t}), ∇ρ(X_{t}, ·)(Y_{t})⟩ \\ = \frac{1}{ρ_{t}^{2}} ∫_{0}^{ρ_{t}} \frac{d}{ds}\{(sψ(Y_{t}) + (ρ_{t} − s)ψ(X_{t}))^{2}⟨Z(γ(s)), γ̇(s)⟩\}ds \\ = \frac{1}{ρ_{t}^{2}} ∫_{0}^{ρ_{t}} (sψ(Y_{t}) + (ρ_{t} − s)ψ(X_{t}))^{2}⟨(∇γ̇Z) ∘ γ, γ̇⟩(s)ds \\ + \frac{2}{ρ_{t}^{2}} ∫_{0}^{ρ_{t}}⟨Z ∘ γ, γ̇⟩(s)(ψ(Y_{t}) − ψ(X_{t}))(sψ(Y_{t}) + (ρ_{t} − s)ψ(X_{t}))ds. \\ ≤ \frac{1}{ρ_{t}^{2}} ∫_{0}^{ρ_{t}} (sψ(Y_{t}) + (ρ_{t} − s)ψ(X_{t}))^{2}⟨(∇γ̇Z) ∘ γ, γ̇⟩(s)ds \\ + 2∥Z∥_{∞}∥ψ∥_{∞}∥∇ψ∥_{∞}ρ_{t}. \end{matrix} (4.9)

Finally, we have ⟨∇ρ(X_{t}, ·)(Y_{t}), P_{X_{t},Y_{t}}Φ_{t}dB̃_{t}⟩ = ⟨P_{Y_{t},X_{t}}∇ρ(X_{t}, ·)(Y_{t}), Φ_{t}dB̃_{t}⟩ =

= −⟨∇ρ(·, Y_{t})(X_{t}), Φ_{t}dB̃_{t}⟩.

Combining this with (4.6), (4.7), (4.8) and (4.9), we arrive at

\begin{matrix}dρ(X_{t}, Y_{t}) ≤ & \sqrt{2} (ψ(X_{t}) − ψ(Y_{t}))⟨∇ρ(·, Y_{t})(X_{t}), Φ_{t}dB̃_{t}⟩ \\ + K_{ψ}ρ(X_{t}, Y_{t})dt + \sqrt{2} ∥ψ∥_{∞}∥β_{t}∥dt. \end{matrix}

Then

M_{t} := \sqrt{2} ∫_{0}^{t} e^{−K_{ψ}s}(ψ(X_{s}) − ψ(Y_{s}))⟨∇ρ(·, Y_{s})(X_{s}), Φ_{s}dB̃_{s}⟩

is a ℚ-martingale such that

ρ(X_{t}, Y_{t}) ≤ e^{K_{ψ}t}M_{t} + \sqrt{2} e^{K_{ψ}t} ∫_{0}^{t} e^{−K_{ψ}s}∥ψ∥_{∞}∥β_{s}∥ds, t ∈ [0, T]. (4.10)

So, by the Doob inequality we obtain

Documenta Mathematica 18 (2013) 297–322

h_{t} := 𝔼_{ℚ} max_{s∈[0,t]} ρ(X_{s}, Y_{s})^{2}

≤ (1 + R)e^{2K_{ψ}t}𝔼_{ℚ} max_{s∈[0,t]} M_{s}^{2}ds

+ 2∥ψ∥_{∞}^{2}(1 + R^{−1})e^{2K_{ψ}t}𝔼_{ℚ}( ∫_{0}^{t} e^{−K_{ψ}s}∥β_{s}∥ds)^{2}

≤ 4(1 + R)e^{2K_{ψ}t} 𝔼_{ℚ}M_{t}^{2} + (1 + R^{−1})∥ψ∥_{∞}^{2} \frac{e^{2K_{ψ}t} − 1}{K_{ψ}} ∫_{0}^{t} 𝔼_{ℚ}∥β_{s}∥^{2}ds

≤ 4(1 + R)∥∇ψ∥_{∞}^{2}e^{2K_{ψ}t} ∫_{0}^{t} e^{−2K_{ψ}s}h_{s}ds

+ (1 + R^{−1})∥ψ∥_{∞}^{2} \frac{e^{2K_{ψ}T} − 1}{K_{ψ}} ∫_{0}^{t} 𝔼_{ℚ}∥β_{s}∥^{2}ds

for any R > 0. Since e^{−2K_{ψ}s} is decreasing in s while h_{s} is increasing in s, by the FKG inequality we have

∫_{0}^{t} e^{−2K_{ψ}s}h_{s}ds ≤ (\frac{1}{t} ∫_{0}^{t} e^{−2Ks}ds) ∫_{0}^{t} h_{s}ds = \frac{1 − e^{−2K_{ψ}t}}{2K_{ψ}t} ∫_{0}^{t} h_{s}ds.

Therefore,

h_{t} ≤ 2(1 + R)∥∇ψ∥_{∞}^{2} \frac{e^{2K_{ψ}T} − 1}{K_{ψ}T} ∫_{0}^{t} h_{s}ds + (1 + R^{−1})∥ψ∥_{∞}^{2} \frac{e^{2K_{ψ}T} − 1}{K_{ψ}} ∫_{0}^{t} 𝔼_{ℚ}∥β_{s}∥^{2}ds

holds for t ∈ [0, T]. Since h_{0} = 0, this implies that

𝔼_{ℚ} max_{t∈[0,T]} ρ(X_{t}, Y_{t})^{2} = h_{T}

≤ (1 + R^{−1})∥ψ∥_{∞}^{2} \frac{e^{2K_{ψ}T} − 1}{K_{ψ}} exp [2(1 + R)∥∇ψ∥_{∞}^{2} \frac{e^{2K_{ψ}T} − 1}{K_{ψ}}]∫_{0}^{T} 𝔼_{ℚ}∥β_{s}∥^{2}ds.

Combining this with the (4.5) and (3.9), we complete the proof.

Theorem 4.2. In the situation of Theorem 4.1,

W_{2,ρ_{∞}}(Π_{μ,ψ}^{T}, Π_{ν,ψ}^{T}) ≤ 2e^{(K_{ψ}+∥∇ψ∥_{∞}^{2})T} W_{2,ρ}(μ, ν), μ, ν ∈ 𝒫(M), T > 0.

Proof. As explained in the proof of “(6) ⇒ (5)”, we only consider μ = δ_{x} and ν = δ_{y}. Let X_{t} solve (4.2) with X_{0} = x, and let Y_{t} solve, instead of (4.4),

d_{I}Y_{t} = \sqrt{2} ψ(Y_{t})P_{X_{t},Y_{t}}Φ_{t}dB̃_{t} + ψ^{2}(Y_{t})Z(Y_{t})dt + N(Y_{t})dl̃_{t}, Y_{0} = y.

Then, repeating the proof of Theorem 4.1, we have, instead of (4.10),

ρ(X_{t}, Y_{t}) ≤ e^{K_{ψ}t}(M_{t} + ρ(x, y)), t ≥ 0  (4.11)

for

M_{t} := \sqrt{2} ∫_{0}^{t} e^{−K_{ψ}s}(ψ(X_{s}) − ψ(Y_{s}))⟨∇ρ(·, Y_{s})(X_{s}), Φ_{s}dB_{s}⟩.

So,

𝔼ρ(X_{t}, Y_{t})^{2} ≤ e^{2K_{ψ}t}{ρ(x, y)^{2} + 2∥∇ψ∥_{∞}^{2} ∫_{0}^{t} e^{−2K_{ψ}s}𝔼ρ(X_{s}, Y_{s})^{2}ds},

which implies

𝔼ρ(X_{t}, Y_{t})^{2} ≤ e^{2(K_{ψ}+∥∇ψ∥_{∞}^{2})t}ρ(x, y)^{2}.

Combining this with (4.11) and the Doob inequality, we arrive at

W_{2,ρ_{∞}}(Π_{x,ψ}^{T}, Π_{y,ψ}^{T})^{2} ≤ 𝔼 max_{t∈[0,T]} ρ(X_{t}, Y_{t})^{2} ≤ e^{2K_{ψ}T} 𝔼 max_{t∈[0,T]}(M_{t} + ρ(x, y))^{2}

≤ 4e^{2K_{ψ}T} 𝔼(M_{T} + ρ(x, y))^{2} = 4e^{2K_{ψ}T} (𝔼M_{T}^{2} + ρ(x, y)^{2})

= 4e^{2K_{ψ}T} (ρ(x, y)^{2} + 2∥∇ψ∥_{∞}^{2} ∫_{0}^{T} e^{−2K_{ψ}t}𝔼ρ(X_{t}, Y_{t})^{2}dt)

≤ 4e^{2(K_{ψ}+∥∇ψ∥_{∞}^{2})T} ρ(x, y)^{2}.

This implies the desired inequality for μ = δ_{x} and ν = δ_{y}.

5 Extensions to non-convex manifolds

As explained in the end of Section 1, combining Theorem 4.1 with a proper conformal change of metric, we are able to establish the following transportationcost inequality on a class of manifolds with non-convex boundary.

Theorem 5.1. Let ∂M ≠ ∅ with 𝕀 ≥ −σ for some constant σ > 0, and let (1.1) hold for some K ∈ ℝ. Then for any f ∈ C_{b}^{∞}(M) with f ≥ 1 and N log f|_{∂M} ≥ σ, and for any μ ∈ 𝒫(M),

W_{2,ρ_{∞}}(F Π_{μ}^{T} , Π_{μ_{F}^{T}}^{T})^{2} ≤ 2∥f∥_{∞}^{2}c(T, f)Π_{μ}^{T} (F log F), F ≥ 0, Π_{μ}^{T} (F) = 1

holds for

c(T, f) = inf_{R>0} \{(1 + R^{−1})\frac{e^{2κ_{f} T} − 1}{κ_{f}} exp [2(1 + R)∥∇f∥_{∞}^{2} \frac{e^{2κ_{f} T} − 1}{κ_{f}}]\},

where

κ_{f} = 5∥f∥_{∞}∥∇f∥_{∞}∥Z∥_{∞} + {3d − 5 + (d − 3)^{+}}∥∇f∥_{∞}^{2} + ∥(Kf^{2} − fΔf)^{+}∥_{∞}.

In particular,

W_{2,ρ_{∞}}(F Π_{o}^{T} , Π_{o}^{T})^{2} ≤ 2∥f∥_{∞}^{2}c(T, f)Π_{o}^{T} (F log F), o ∈ M, F ≥ 0, Π_{o}^{T} (F) = 1.

Proof. Let f ∈ C_{b}^{∞}(M) such that f ≥ 1. Since 𝕀 ≥ −σ and N log f|_{∂M} ≥ σ, by [21, Lemma 2.1] the boundary ∂M is convex under the new metric

⟨·, ·⟩′ = f^{−2}⟨·, ·⟩.

Let Δ′ and ∇′ be induced by the new metric. Then (see formula (2.2) in [14])

L = f^{−2}(Δ′ + Z′), Z′ := f^{2}Z + \frac{d − 2}{2}∇f^{2}.

Let Ric′ be the Ricci curvature induced by the new metric, we have (cf. formula (3.2) in [6])

Ric′ = Ric + (d − 2)f^{−1}Hess_{f} + (f^{−1}Δf − (d − 3)|∇ log f|^{2})⟨·, ·⟩. (5.1)

Since the Levi-Civita connection induced by ⟨·, ·⟩′ satisfies (cf. [3, Theorem 1.59(a)])

∇′_{U}V = ∇_{U}V − ⟨U, ∇ log f⟩V − ⟨V, ∇ log f⟩U + ⟨U, V ⟩∇ log f, U, V ∈ T M,

we have

⟨∇′_{U}Z′, U⟩′ = f^{−2}{⟨∇_{U}Z′, U⟩ − ⟨Z′, ∇ log f⟩|U|^{2}}

= 2⟨U, ∇ log f⟩⟨Z, U⟩ + ⟨∇_{U}Z, U⟩ + \frac{d − 2}{2f^{2}}Hess_{f^{2}}(U, U)

− ⟨Z, ∇ log f⟩|U|^{2} − \frac{d − 2}{2}⟨∇ log f^{2}, ∇ log f⟩|U|^{2}

≤ ⟨∇_{U}Z, U⟩ + 3|∇ log f| · |Z| · |U|^{2} + (d − 2)f^{−1}Hess_{f}(U, U).

Combining this with (5.1), we obtain

Ric′(U, U) − ⟨∇_{U}Z′, U⟩′

≥ Ric(U, U) − ⟨∇_{U}Z, U⟩ + {f^{−1}Δf − (d − 3)|∇ log f| − 3|Z| · |∇ log f|}|U|^{2} ≥ −K′⟨U, U⟩′, U ∈ T M,

where

K′ = sup_{M}{Kf^{2} − fΔf + (d − 3)|∇f|^{2} + 3|Z|f|∇f|}.  (5.2)

Noting that f ≥ 1, we have

\begin{matrix}\sqrt{⟨Z′, Z′⟩′} = f^{−1}|f^{2}Z + (d − 2)f∇f| ≤ ∥f∥_{∞}∥Z∥_{∞} + (d − 2)∥∇f∥_{∞}, \\ \sqrt{⟨∇′f^{−1}, ∇′f^{−1}⟩′} = f|∇f^{−1}| ≤ ∥∇f∥_{∞}. \end{matrix} (5.3)

Documenta Mathematica 18 (2013) 297–322

Letting K_{ψ} be defined in Theorem 4.1 for the manifold (M, ⟨·, ·⟩′) and L = ψ^{2}(Δ′ + Z′) with ψ = f^{−1}, we deduce from f ≥ 1, (5.2) and (5.3) that

K_{ψ} ≤ κ_{f}.

Therefore, C(T, ψ) ≤ c(T, f) and thus, Theorem 4.1 implies

W_{2,ρ′_{∞}}(F Π_{μ}^{T} , Π_{μ_{F}^{T}}^{T})^{2} ≤ 2c(T, f)Π_{μ}^{T} (F log F), F ≥ 0, Π_{μ}^{T} (F) = 1,

where ρ′_{∞} is the uniform distance on M^{T} induced by the metric ⟨·, ·⟩′. The proof is completed by noting that ρ_{∞} ≤ ∥f∥_{∞}ρ′_{∞}.

Similarly, since K_{ψ} ≤ κ_{f} and

ρ′ ≤ ρ ≤ ∥f∥_{∞}ρ,

the following result from Theorem 4.2 by taking ψ = f^{−1}.

Theorem 5.2. In the situation of Theorem 5.1,

W_{2,ρ_{∞}}(Π_{μ}^{T} , Π_{ν}^{T}) ≤ 2∥f∥_{∞}e^{(κ_{f} +∥∇f^{−1}∥_{∞}^{2})T} W_{2,ρ}(μ, ν), μ, ν ∈ 𝒫(M), T > 0.

As a consequence of Theorems 5.1 and 5.2, we present below an explicit transportation-cost inequalities for a class of non-convex manifolds.

Corollary 5.3. Assume that (1.1) holds for some K ≥ 0 and the injectivity radius i_{∂M} of ∂M is strictly positive. Let σ ≥ 0 and γ, k, > 0 be such that −σ ≤ 𝕀 ≤ γ and Sect_{M} ≤ k. Let

0 < r ≤ min \{i_{∂M}, \frac{1}{\sqrt{k}} arcsin (\begin{matrix}√ & \overline{k} \\ √ & \overline{k + γ^{2}} \end{matrix} )\}.

(i) The transportation-cost inequality

W_{2,ρ_{∞}}(F Π_{μ}^{T} , Π_{μ_{F}^{T}}^{T})^{2} ≤ (2 + rdσ)^{2} \frac{e^{2θT} − 1}{θ} exp [\frac{4(e^{2θT} − 1)}{θ}]Π_{μ}^{T} (F log F)

holds for all μ ∈ 𝒫(M) and F ≥ 0 with Π_{μ}^{T} (F) = 1, where

θ = K(1+rdσ+ \frac{r^{2}d^{2}σ^{2}}{4} )+ \frac{dσ}{r} (3d−5+(d−3)^{+} + \frac{d^{2}}{2} )σ^{2} +5∥Z∥_{∞}σ(1+ \frac{rdσ}{2} ).

In particular,

W_{2,ρ_{∞}}(F Π_{o}^{T} , Π_{o}^{T})^{2} ≤ (2 + rdσ)^{2} \frac{e^{2θT} − 1}{θ} exp [\frac{4(e^{2θT} − 1)}{θ}]Π_{o}^{T} (F log F)

holds for all F ≥ 0 with Π_{μ}^{T} (F) = 1. (ii) For any T > 0 and μ, ν ∈ 𝒫(M),

W_{2,ρ_{∞}}(Π_{μ}^{T} , Π_{ν}^{T}) ≤ (2 + σrd)e^{(θ+σ^{2})T} W_{2,ρ}(μ, ν).

Proof. Let

h(s) = cos (\sqrt{k} s) − \frac{γ}{\sqrt{k}} sin (\sqrt{k} s), s ≥ 0.

Then h is the unique solution to the equation

h′′ + kh = 0, h(0) = 1, h′(0) = −γ.

Up to an approximation argument presented in the proof of [20, Theorem 1.1], we may apply Theorem 5.1 to

f = 1 + σφ ∘ ρ_{∂M},

where ρ_{∂} is the Riemannian distance to ∂M, which is smooth on {ρ_{∂M} < i_{∂M}}, and

\begin{matrix}α = (1 − h(r))^{1−d} ∫_{0}^{r}(h(s) − h(r))^{d−1}ds, \\ φ(s) = \frac{1}{α} ∫_{0}^{s}(h(t) − h(r))^{1−d}dt ∫_{t∧r}^{r}(h(u) − h(r))^{d−1}du, s ≥ 0. \end{matrix}

We have φ(0) = 1, 0 ≤ φ′ ≤ φ′(0) = 1. Moreover, as observed in [20, Proof of Theorem 1.1],

α ≥ \frac{r}{d}, φ(r) ≤ \frac{r^{2}}{2α} ≤ \frac{dr}{2} , Δφ ∘ ρ_{∂M} ≥ −\frac{1}{α} ≥ −\frac{d}{r}.

So,

∥f∥_{∞} ≤ 1 + σφ(r) ≤ 1 + \frac{rdσ}{2} , ∥∇f∥_{∞} ≤ φ′(0) = σ, Δf ≥ −\frac{σd}{r}. (5.4)

Noting that (recall that K ≥ 0)

sup(Kf^{2}) ≤ K(1 + rdσ + \frac{r^{2}d^{2}σ^{2}}{4} ),

from (5.4) we conclude that κ_{f} ≤ θ. So, (i) follows from (1.5) and 5.1 for R = 1, and (ii) follows from Theorem 4.2 and (5.4).

Acknowledgement. The author would like to thank the referee for very helpful comments and corrections.

References

[1] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below, Bull. Sci. Math. 130(2006), 223–233.

[2] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds, Stoch. Proc. Appl. 2009.

[3] A. L. Besse, Einstein Manifolds, Springer, Berlin, 1987.

[4] S. Bobkov, I. Gentil, M. Ledoux, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pure Appl. 80 (2001), 669–696.

[5] H. Djellout, A. Guillin, L.-M. Wu, Transportation cost information inequalities and applications to random dynamical systems and diffusions, Ann. Probab. 32 (2004), 2702–2732.

[6] S. Fang, F.-Y. Wang, B. Wu, Transportation-cost inequality on path spaces with uniform distance, Stoch. Proc. Appl. 118(2008), 2181–197.

[7] D. Feyel, A. Üstünel, Measure transport on Wiener space and the Girsanov theorem, C.R. Acad. Paris 334(2002), 1025–1028.

[8] N. Ikeda, S. Watanabe, Stochastic Differential Equations, North-Holland, New York, 1989.

[9] W.S. Kendall, Nonnegative Ricci curvature and the Brownian coupling property, Stochastics 19 (1986), 111C 129.

[10] M. Ledoux, The geometry of Markov diffusion generators, Ann. Facul. Sci. Toulouse 9(2000), 305–366.

[11] M.-K. von Reness, K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Math. 58(2005), 923–940.

[12] F. Otto, C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173(2000), 361– 400.

[13] M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal. 6(1996), 587–600.

[14] A. Thalmaier, F.-Y. Wang, Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. Funct. Anal. 155:1(1998),109–124.

[15] F.-Y. Wang, Application of coupling methods to the Neumann eigenvalue problem, Probab. Theory Related Fields 98 (1994), 299–306.

[16] F.-Y. Wang,On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups, Probability Theory Related Fields 108(1997), 87–101.

[17] F.-Y. Wang, Transportation cost inequalities on path spaces over Riemannian manifolds, Illinois J. Math. 46 (2002), 1197–1206.

[18] F.-Y. Wang, Probability distance inequalities on Riemannian manifolds and path spaces, J. Funct. Anal. 206 (2004), 167–190.

[19] F.-Y. Wang, Equivalence of dimension-free Harnack inequality and curvature condition, Integral Equation and Operator Theory 48(2004), 547–552.

[20] F.-Y. Wang, Gradient estimates and the first Neumann eigenvalue on manifolds with boundary, Stoch. Proc. Appl. 115(2005), 1475–1486.

[21] F.-Y. Wang, Estimates of the first Neumann eigenvalue and the log-Sobolev constant on nonconvex manifolds, Math. Nachr. 280(2007), 1431–1439.

[22] F.-Y. Wang, Second fundamental form and gradient of Neumann semigroups, J. Funct. Anal. 256(2009), 3461–3469.

[23] F.-Y. Wang, Harnack inequalities on manifolds with boundary and applications, J. Math. Pures Appl. 94(2010), 304–321.

[24] F.-Y. Wang, Semigroup properties for the second fundamental form, Docum. Math. 15(2010), 543–559.

[25] F.-Y. Wang, Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures, Science in China (Mathematics) 53(2010), 895– 904.

[26] F.-Y. Wang, Analysis on path space over Riemannian manifolds with boundary, Comm. Math. Sci. 9(2011), 1203–1213.

[27] L.-M. Wu, Z.-L. Zhang, Talagrand’s T_{2}-transportation inequality w.r.t. a uniform metric for diffusions, Acta Math. Appl. Sin. Engl. Ser. 20 (2004), 357–364.

Feng-Yu Wang

School of Mathematical Sci. and Lab. Math. Com. Sys. Beijing Normal University Beijing 100875 China wangfy@bnu.edu.cn; and Department of Mathematics Swansea University Singleton Park, SA2 8PP UK F.Y.Wang@swansea.ac.uk