Functional inequalities on manifolds with non-convex boundary

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Functional inequalities on manifolds with non-convex boundary

Li-Juan Cheng

, Anton Thalmaier

& James Thompson

1Mathematics Research Unit, FSTC, University of Luxembourg Maison du Nombre, 6, Avenue de la Fonte,

4364 Esch-sur-Alzette, Grand Duchy of Luxembourg;

2Department of Applied Mathematics, Zhejiang University of Technology Hangzhou 310023, The People’s Republic of China

Email: [email protected], [email protected], [email protected], [email protected]

Abstract In this article, new curvature conditions are introduced to establish functional inequalities includ- ing gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non-convex boundary.

Keywords Ricci curvature, gradient inequality, log-Sobolev inequality, geometric flow MSC(2010) 60J60, 58J65, 53C44

Citation: L.-J. Cheng, A. Thalmaier and J. Thompson. Functional inequalities on manifolds with non-convex boundary. https://doi.org/10.1007/s11425-000-0000-0

1 Introduction

Let (M, g) be a complete and connected Riemannian manifold of dimension d > 2, with Riemannian distance ρ, boundary ∂M and inward pointing unit normal vector N. Define the second fundamental form of the boundary by

II(X, Y) =− h∇XN, Yi, X, Y ∈Tx∂M, x∈∂M

whereT ∂M denotes the tangent bundle of∂M. In order to study non-convex boundaries we will perform a conformal change of metric such that the boundary is convex under the new metric. In particular, we will use the fact that if

D:={φ∈C_b²(M) : infφ= 1,II>−Nlogφ}

andφ∈D then the boundary∂M is convex under the metricφ⁻²g(see [16, Theorem 1.2.5]).

Given aC¹-vector field Z onM, consider the elliptic operatorL:= ∆ +Z and letX_t^x be a reflecting L-diffusion process starting from X₀^x=x. ThenX_t^xsolves the Stratonovich equation

dX_t^x=√

2u^x_t ◦dBt+Z(X_t^x) dt+N(X_t^x) dl^x_t, X₀^x=x

* Corresponding author

where u^x_t is the horizontal lift of X_t^x to the orthonormal frame bundle O(M) with π(u^x₀) = x, B_t is a standard R^d-valued Brownian motion defined on a complete naturally filtered probability space (Ω,{Ft}t>0,P) and l^x_t is a continuous adapted nondecreasing and nonnegative process which increases only on{t>0 : X_t^x∈∂M}. The processl_t^xis the local time ofX_t^xon∂M.

We assume that X_t^x is non-explosive for each x ∈ M. Then the diffusion process X_t^x gives rise to the Neumann semigroupPtwhich solves the diffusion equation (∂t−L)Pt= 0 with Neumann boundary conditionN Pt= 0. FurthermorePtf(x) =E[f(X_t^x)] for eachf ∈Cb(M) .

In [7], Hsu found a probabilistic formula for ∇P_tf for compact manifolds with boundary, which he used to derive a gradient estimate. Feng-Yu Wang extended it to the non-compact case [16, Theorem 3.2.1] under the assumption that |∇P.f| is uniformly bounded on [0, t]×M. Wang’s formula is given below by Theorem 2.1. In [16, Proposition3.2.7], he proved that if

Ric^Z := Ric− ∇Z>K for someK∈C(M) and if there existsφ∈ Dsuch that

K˜_φ:= inf

M

φ²K+1

2Lφ²− |∇φ²| |Z| −(d−2)|∇φ|²

>−∞ (1.1)

then|∇P.f|is uniformly bounded on [0, t]×M by an expression involving the constant ˜Kφ.

In this article, we revisit this problem using coupling methods. In particular, we prove (see Theo- rem 2.2) that if there existsφ∈ Dand a constantK_φ such that

Ric^Z+Llogφ−2|∇logφ|²>K_φ

then|∇P.f|is uniformly bounded on [0, t]×M and our upper bound improves that of [16, Proposition 3.2.7]. We construct a suitable function φ in Proposition 3.3, under the assumption that there exist non-negative constantsσand θsuch that−σ6II6θand a positive constantr0 such that on∂r₀M :=

{x∈M:ρ_∂(x)6r₀}the functionρ_∂ is smooth, the norm ofZis bounded and Sect6kfor some positive constantk.

F.-Y. Wang also considered Harnack and transportation-cost inequalities on manifolds with boundary [16]. We reconsider these problems too and find that the curvature conditions used to establish these inequalities can also be weakened and simplified. It is worth mentioning that we find a transportation- cost inequality on the path space of the reflecting diffusion process which (see Theorem 2.8) recovers the results for the convex boundary case, making this aspect of the theory of functional inequalities on path space complete.

Let us now describe the organization of this article. In Section 2, we prove the gradient estimates, Har- nack inequalities and transportation-cost inequalities for the Neumann semigroup via coupling methods.

In Section 3, we construct a functionφwhich satisfies the new curvature conditions.

2 Functional inequalities

2.1 Gradient estimates

A derivative formula forPtf that does not involve derivatives off is typically called a Bismut formula (see [4, 5]). The Bismut formula we introduce is of a type due originally to Thalmaier [9]. As mentioned in the introduction, Hsu [7] found this type of formula for compact manifolds with boundary. The following formula for manifolds with boundary, due to F.-Y. Wang [16, Theorem 3.2.1], does not require compactness. See also [1] for recent work on probabilistic representations of the derivative of Neumann semigroups.

Theorem 2.1. Let t >0andu0∈Ox(M)be fixed. SupposeK∈C(M)andσ∈C(∂M)are such that Ric_Z >K andII>σ. Assume that

sup

s∈[0,t]

E^x

exp

− Z s

0

K(Xr) dr− Z s

0

σ(Xr) dlr

<∞.

Then there exists a progressively measurable process{Q_s}_s∈[0,t] onR^d⊗R^d such that

Q₀=I, kQsk6exp

− Z s

0

K(X_r) dr− Z s

0

σ(X_r) dl_r

, s∈[0, t]

and for any f ∈C_b¹(M)such that ∇P.f is bounded on [0, t]×M, for any h∈C¹([0, t]) with h(0) = 0 andh(t) = 1, we have

u⁻¹₀ ∇Ptf(x) =E^x

Qtu⁻¹_t ∇f(Xt)

= 1

√ 2E^x

f(Xt)

Z t 0

h(s)Q˙ sdBs

.

In order to use this formula it is necessary to check the uniform boundedness of ∇P.f on [0, t]×M. In [16, Proposition 3.2.7], F.-Y. Wang did so using a conformal change of metric such that under the new metric the boundary is convex, and by then making a time change of theL-diffusion processXt. Here, we use coupling methods to study this problem again and obtain improved upper bounds.

Theorem 2.2. If there exist φ∈D and a constant Kφ such that

Ric^Z+Llogφ−2|∇logφ|²>Kφ (2.1) then for allf ∈C¹(M)such that f is constant outside a compact set,

|∇Ptf|6kφk_∞k∇fk_∞e^−Kφt, t >0.

Proof. We start with a conformal change of the metric g. Since φ∈D, the boundary∂M is convex under the metricg⁰:=φ⁻²g. Let ∆⁰ and∇⁰ be the Laplacian and gradient operator associated with the metricg⁰. Then

L=φ⁻² ∆⁰+φ²(Z+ (d−2)∇logφ)

=φ⁻²(∆⁰+Z⁰) (2.2)

whereZ⁰ :=φ²(Z+ (d−2)∇logφ). For the processXtgenerated byL, viewed as a process on (M, g⁰), denoting by d_I the Itˆo differential, it follows that

dIXt=√

2φ⁻¹(Xt)utdBt+φ⁻²(Xt)Z⁰(Xt) dt+N⁰(Xt) dlt, X0=x (2.3) whereB_tis the Brownian motion and the liftu_tand boundary local timel_tare defined now with respect to the metricg⁰. Recall that in local coordinates, the Itˆo differential of a continuous semimartingale Xt

onM is given (see [6] or [2]) by

(dIXt)^k = dX_t^k+1 2

d

X

i,j=1

Γ^0k_ij(Xt) dhXⁱ, X^jit, 16k6d

where Γ^0k_ij are the Christoffel symbols ofg⁰. Similarly, letY_tsolve d_IY_t=√

2φ⁻¹(Y_t)(1_{{(Xt,Yt)/∈cut}}P_X⁰

t,Y_tu_tdB_t+ 1_{{(Xt,Yt)∈cut}}u˜_tdB_t⁰) +φ⁻²(Y_t)Z⁰(Y_t) dt+N⁰(Y_t) d˜l_t with Y₀ =y, lift ˜u_t and boundary local time ˜l_t, where cut⊂M ×M denotes the set of cut points and where B_t⁰ is a Brownian motion independent of B_t (see [16, Theorem 3.2.5] or [11, Section 2.1]). Now, for (x, y)∈/ cut andx6=y, define

I_Z^φ(x, y) :=

d

X

i=1

(U_i)²ρ⁰(x, y) +

φ⁻²(y)Z⁰(y),∇⁰ρ⁰(x,·)(y)0

+

φ⁻²(x)Z⁰(x),∇⁰ρ⁰(·, y)(x)0

where{U_i}^d_i=1are vector fields onM ×M such that∇⁰U_i(x, y) = 0 and Ui(x, y) = (φ⁻¹(x)Vi, φ⁻¹(y)P_x,y⁰ Vi), 16i6d

for {V_i}^d_i=1 a g⁰-orthonormal basis of T_xM. Here P_x,y⁰ denotes parallel displacement from x to y with respect to the metric g⁰. Denote byρ⁰ the distance function for the metricg⁰. Since the boundary∂M is convex underg⁰, that isN⁰ρ⁰|∂M 60, by a slight modification of the proof of [11, Theorem 2.1.6], we have the following result which is similar to [16, Theorem 3.2.5]: if there existsJ ∈C(M×M) such that J >I_Z^φ outside the cut locus andD(M), where D(M) :={(x, x) : x∈M}, then

dρ⁰(X_t, Y_t)6√

2 φ⁻¹(X_t)−φ⁻¹(Y_t)

db_t+J(X_t, Y_t) dt (2.4) up to the coupling timeτ:= inf{t>0 : Xt=Yt}, wherebtis a one-dimensional Brownian motion. From this, it now suffices for us to estimate the termI_Z^φ. Writeρ⁰ =ρ⁰(x, y) and for a minimizing g⁰-geodesic γ withγ(0) =xandγ(ρ⁰) =y let

J_i(s) =φ⁻¹(γ(s))P_γ(0),γ(s)⁰ V_i, 16i6d

where Ji(0) = φ⁻¹(x)Vi and Ji(ρ⁰) =φ⁻¹(y)P_x,y⁰ Vi. Since P_γ(0),γ(s)⁰ Vi are parallel vector fields along γ with respect to the metricg⁰, we have that for (x, y)∈/cut∪D(M), that

d

X

i=1

(Ui)²ρ⁰(x, y)

6

d

X

i=1

Z ρ⁰ 0

n|∇⁰_γ˙Ji|⁰²− hR⁰( ˙γ, Ji)Ji,γi˙ ⁰o (s) ds

=d Z ρ⁰

0

φ⁻²(γ(s))h∇logφ(γ(s)),γ(s)i˙ ²ds− Z ρ⁰

0

φ⁻²(γ(s))Ric⁰( ˙γ(s),γ(s)) ds.˙ (2.5) On the other hand

φ⁻²(x)hZ⁰(x),∇⁰ρ⁰(·, y)(x)i⁰+φ⁻²(y)hZ⁰(y),∇⁰ρ⁰(x,·)(y)i⁰

= Z ρ⁰

0

d ds

n

φ⁻²(γ(s))hZ⁰(γ(s)),γ(s)i˙ ⁰o ds

= Z ρ⁰

0

φ⁻²(γ(s))

(∇⁰_γ˙Z⁰)◦γ,γ˙⁰ (s) ds

−2 Z ρ⁰

0

φ⁻²(γ(s))h∇logφ(γ(s)),γ(s)i hZ˙ ⁰(γ(s)),γ(s)i˙ ⁰ ds. (2.6) Moreover

hZ⁰(γ(s)),γ(s)i˙ ⁰=hZ,γ(s)i˙ + (d−2)h∇logφ,γ(s)i.˙ Combining this with (2.5) and (2.6), we have

I_Z^φ(x, y)6− Z ρ⁰

0

φ⁻²(γ(s))((Ric^Z)⁰( ˙γ(s),γ(s)) + (d˙ −4)h∇logφ,γ(s)i˙ ²ds

− Z ρ⁰

0

2h∇logφ,γ(s)ihZ,˙ γ(s)i) ds.˙ (2.7) By [3, Theorem 1.159], in which the Laplacian differs from our’s by a negative sign, we know that

(Ric^Z)⁰( ˙γ,γ) = Ric˙ ⁰( ˙γ,γ)˙ − h∇⁰_γ˙Z⁰,γi˙ ⁰

= Ric^Z( ˙γ,γ) +˙ 1

2Lφ²−2h∇logφ,γi hZ,˙ γi −˙ (d−2)hγ,˙ ∇logφi²−2|∇φ|² and, noting that|γ|˙ =φ, we thus have

(Ric^Z)⁰( ˙γ(s),γ(s)) + (d˙ −4)h∇logφ,γ(s)i˙ ²+ 2h∇logφ,γ(s)ihZ,˙ γ(s)i˙

= Ric^Z( ˙γ(s),γ(s)) +˙ 1

2Lφ²−2hγ,˙ ∇logφi²−2|∇φ|²

> Ric^Z( ˙γ(s),γ(s)) +˙ 1

2Lφ²−4|∇φ|²

= Ric^Z( ˙γ(s),γ(s)) +˙ φ²Llogφ−2|∇φ|². (2.8) Consequently, using the condition (2.1), letting Xt=Yt after coupling time, and then combining (2.7) with (2.8) and (2.4), we arrive at

dρ⁰(X_t, Y_t)6√

2(φ⁻¹(X_t)−φ⁻¹(Y_t)) db_t−K_φρ⁰(X_t, Y_t) dt. (2.9) From this, we know that

E^(x,y)[ρ⁰(X_t, Y_t)]6e^−Kφtρ⁰(x, y).

Then, observing thatρ⁰ 6ρ6kφk_∞ρ⁰, we have

|∇P_tf|(x) = lim

y→x

Ptf(x)−Ptf(y) ρ(x, y)

= lim

y→x

E^(x,y)

f(Xt)−f(Yt) ρ(X_t, Y_t)

ρ(Xt, Yt) ρ⁰(X_t, Y_t)

ρ⁰(Xt, Yt) ρ⁰(x, y)

ρ⁰(x, y) ρ(x, y)

6kφk_∞k∇fk_∞e^−Kφt

which completes the proof.

Remark 2.3. (i) Since (U_d)²ρ⁰6= 0, it was indeed necessary to account for this quantity in inequality (2.5), correcting the proof of [16, Theorem 3.4.6].

(ii) Compared with the proof of [16, Theorem 3.4.6], our choice of vector fieldJi yields a simpler result.

(iii) In [17], a certain technical assumption which was used to ensure the uniformly boundedness of

|∇P.f|on [0, t]×M is no longer needed in the results.

The following results remove the additional condition in [16, Corollary 3.6.5 (1)] and [17, Corollary 1.2 (1)] to ensure the uniform boundedness of|∇P.f|on [0, t]×M and give a another proof of an extension of these inequalities toL^p forms forp >1:

Theorem 2.4. If there exists φ∈D such that for p >1 the inequality

Ric^Z+Llogφ−p|∇logφ|²>K_φ,p (2.10) holds, then fort >0 andf ∈C_b¹(M),

|∇Ptf|6 1

φe^−Kφ,pt

P_t(φ|∇f|)^{p/(p−1)(p−1)/p} .

Proof. The lower bound (2.10) implies Ric^Z+Llogφ−2|∇logφ|²is bounded below (sinceφ∈Dimplies

|∇logφ|bounded). By Theorem 2.2, it follows that|∇P.f| is bounded on [0, t]×M. Furthermore Ric^Z >Kφ,p−Llogφ+p|∇logφ|²=Kφ,p+1

pφ^pLφ^−p and II>−Nlogφ and so, by Theorem 2.1, there exists{Qs}_s∈[0,t] such that

kQtk6exp

−Kφ,pt−1 p

Z t 0

φ^pLφ^−p(Xs) ds+ Z t

0

Nlogφ(Xs) dls

(2.11) with

|∇Ptf|^p6

Pt(φ|∇f|)^p/(p−1)p−1

E

φ^−p(Xt)kQtk^p

. (2.12)

It therefore suffices to give the upper bound estimate of the following term:

E

φ^−p(Xt) exp

− Z t

0

φ^pLφ^−p(Xs) ds+p Z t

0

Nlogφ(Xs) dls

.

To this end, by the Itˆo formula, it is easy to see that

dφ^−p(X_t) =h∇φ^−p(X_t), u_tdB_ti+Lφ^−p(X_t) dt+N φ^−p(X_t) dl_t

=h∇φ^−p(Xt), utdBti −pφ^−p(Xt)

−1

pφ^pLφ^−p(Xt) dt+Nlogφ(Xt) dlt

.

So

Mt=φ^−p(Xt) exp

− Z t

0

φ^p(Xs)Lφ^−p(Xs) ds+p Z t

0

Nlogφ(Xs) dls

is a positive local martingale. Thus E

φ^−p(Xt) exp

− Z t

0

φ^p(Xs)Lφ^−p(Xs) ds+p Z t

0

Nlogφ(Xs) dls

6φ^−p(x).

Combining this with (2.11) and (2.12) completes the proof.

Corollary 2.5. If there exists φ∈D such that for p >1 the inequality Ric^Z+Llogφ−p|∇logφ|²>K_φ,p holds, then fort >0 andf ∈C_b¹(M),

|∇P_tf|6kφk_∞e^−Kφ,pt(P_t|∇f|^p/(p−1))^(p−1)/p; and forf ∈Bb(M)andt >0,

|∇Ptf|²6kφk²_∞ Kφ,2

e^2Kφ,2t−1Ptf². (2.13)

Proof. The first assertion follows from Theorem 2.4 by observingφ>1. As Theorem 2.1 can be used under our condition directly, the main idea of the proof of (2.13) is similar to that of [16, Corollary 3.2.8], so we skip it here.

Note that taking the limitp↓1 in Corollary 2.5 improves Theorem 2.2 by replacing the constantK_φ withKφ,1.

2.2 Harnack inequalities

In [16, Theorem 3.4.7] and [14, Theorem 3.1], F.-Y. Wang used a coupling method to obtain dimension free Harnack inequalities and a log-Harnack inequality on manifolds with boundary, assuming Ric^Z >K for some K ∈C(M) withφ∈D such that ˜K_φ is finite (where the quantity ˜K_φ is defined as in (1.1)).

The coefficient involved in these inequalities is:

2 ˜K_φ⁻+ 4kφZ+ (d−2)∇φk_∞k∇logφk_∞+ 2dk∇logφk²_∞.

We now give the following result, weakening the curvature condition, in terms of a different coefficient:

Theorem 2.6. Assume there existsφ∈ D such that

Ric^Z+Llogφ−3|∇logφ|²>K_φ,3 (2.14) for some constant K_φ,3. Then forT >0,x, y∈M,p >kφk²_∞ andf ∈C_b¹(M), we have

P_Tf(y)^p

6P_Tf^p(x) exp √

p(√

p−1)K_φ,3kφk²_∞ρ²(x, y) 8δ_p(√

p−1−δ_p)(e^2Kφ,3T−1)

,

whereδ_p= maxn

kφk∞−1,

√p−1 2

o.

Proof. Fixx, y∈M andT >0. As in the proof of Theorem 2.2, we consider the processX_tgenerated by L= ∆ +Z under the metric g⁰ :=φ⁻²g, for which the boundary (∂M, g⁰) is convex. LetXt solve equation (2.3) withX0 =x. For a strictly positive functionξ∈C([0, T)), to be later determined, letYt

solve

d_IY_t=√

2φ⁻¹(Y_t) 1_{{(Xt,Yt)/∈cut}}P_X⁰

t,Y_tu_tdB_t+√

2φ⁻¹(Y_t)1_{{(Xt,Yt)∈cut}}˜u_tdB_t⁰ +φ⁻²_t Z⁰(Y_t) dt−φ⁻¹(Y_t)ρ⁰(X_t, Y_t)

φ⁻¹(Xt)ξ(t) ∇⁰ρ⁰(X_t,·)(Yt) dt+N⁰(Y_t) d˜l_t

for t ∈[0, T), with Y0 =y, where ˜lt is the local time of Yt on∂M, ˜ut is the lift process andB⁰_t is the Brownian motion independent ofBt from earlier. In the following, we begin with the same argument as in the proof of [16, Theorem 3.4.7]. The different part is how to use our curvature condition to get a new estimate for the radial processρ⁰(X_t, Y_t). To this end, as explained in the proof of Theorem 2.2, we may for the sake of conciseness disregard certain technical considerations relating to the cut locus ofM. Consider the process (Xt, Yt) starting from (x, y), which is a well defined continuous process fort6T∧ζ whereζ is the explosion time ofY_t; that isζ:= lim_n→∞ζ_n forζ_n := inf{t >0 :ρ⁰(y, Y_t)>n}. Let

d ˜Bt= dBt+ ρ⁰(Xt, Yt)

√2ξ(t)φ⁻¹(Xt)u⁻¹_t ∇⁰ρ⁰(·, Yt)(Xt) dt, 06t < T∧ζ. (2.15) By the Girsanov theorem, for anys∈(0, T) the processBt is ad-dimensional Brownian motion under the probability measureR_sPfor

R_s:= exp

− Z s

0

ρ⁰(Xt, Yt)

ξ(t)φ⁻¹(X_t)h∇⁰ρ(·, Y_t)(X_t), u_tdB_ti⁰−1 2

Z s 0

ρ⁰(Xt, Yt)² ξ²φ⁻²(X_t)dt

.

LetQ=R_T∧ζP. It has been shown in the proof of [16, Theorem 3.4.7] thatQ(ζ=T) = 1 and the coupling is successful up to the timeT. In the following, we look at the processes under the new measureQ. Then

dIXt=√

2φ⁻¹(Xt)utd ˜Bt+φ⁻²(Xt)Z⁰(Xt) dt−ρ⁰(Xt, Yt)

ξ(t) ∇⁰ρ⁰(Xt,·)(Yt) dt+N⁰(Xt) d˜lt; d_IY_t=√

2φ⁻¹(Y_t)P_X⁰

t,Y_tu_td ˜B_t+φ⁻²(Y_t)Z⁰(Y_t) dt+N⁰(Y_t) d˜l_t, t6T.

Since

Ric^Z+Llogφ−2|∇logφ|²>Kφ,3+|∇logφ|², t∈[0, T], by a similar calculation as for (2.8) and (2.9) we find

dρ⁰(Xt, Yt)6√

2(φ⁻¹(Xt)−φ⁻¹(Yt))D

∇⁰ρ⁰(·, Yt)(Xt), utd ˜Bt

E0

−

Z ρ⁰(Xt,Yt) 0

(K_φ,3+|∇logφ|²)(γ(s)) ds

!

dt−ρ⁰(X_t, Y_t)

ξ(t) dt, 06t < T (2.16) which implies

dρ⁰(X_t, Y_t)² ξ(t) 6 2√

2

ξ(t)ρ⁰(X_t, Y_t) φ⁻¹(X_t)−φ⁻¹(Y_t)D

∇⁰ρ⁰(·, Yt)(X_t), u_td ˜B_tE⁰

−ρ⁰(Xt, Yt)² ξ²(t)

ξ(t) + 2K˙ φ,3ξ(t) + 2

dt, 06t < T, (2.17) since the positive term coming from the covariation is smaller than the opposite of the negative term involving|∇logφ|². Now forθ∈(0,2) let

ξ(t) = (2−θ) Z T

t

e^{−2Kφ,3(t−s)} ds, t∈[0, T)

so thatξsolves the equation

ξ(t) + 2K˙ φ,3ξ(t) + 2 =θ, t∈[0, T).

Combining this with (2.17), we find dρ⁰(Xt, Yt)²

ξ(t) 6 2√ 2

ξ(t)ρ⁰(Xt, Yt) φ⁻¹(Xt)−φ⁻¹(Yt)D

∇⁰ρ⁰(·, Yt)(Xt), utd ˜Bt

E0

−ρ⁰(Xt, Yt)² ξ(t)² θdt.

The remainder of argument is given by the proof of [16, Theorem 3.4.7].

2.3 Transportation-cost inequalities

Consider µ, ν ∈ P(M) where P(M) denotes the space of all probability measures on M. Recall the L^p-Wasserstein distance betweenµandν is

W_p(µ, ν) = inf

η∈C(µ,ν)

Z

M×M

ρ(x, y)^pdη(x, y) 1/p

whereC(µ, ν) is the set for couplings ofµandν. When the manifold has no boundary, it is well known that the curvature condition,

Ric^Z >K for some constantK is equivalent to

W_p(µP_t, νP_t)6W_p(µ, ν) e^−Kt, µ, ν ∈P(M),

whereµP_t∈P(M) is defined by (µP_t)(A) =µ(P_t1_A) for measurable setA. This equivalence is due to [10]

which is extended to the manifolds with convex boundary [15]. Using a coupling method, with the effect of the cut locus accommodated as in the proof of Theorem 2.6, we obtain the following transportation-cost inequality.

Theorem 2.7. If there exists φ∈D and a constant Kφ,3 satisfying

Ric^Z+Llogφ−3|∇logφ|²>Kφ,3

then

W2(µPt, νPt)6kφk∞e^−Kφ,3tW2(µ, ν).

Proof. By [16, Theorem 4.4.2], it suffices to only consider µ = δ_x and ν = δ_y. Let φ be a smooth function inD and recall thatL=φ⁻²(∆⁰+Z⁰) for the manifold (M, g⁰) as in (2.2), where g⁰ =φ⁻²g.

LetXtandYtsolve the following SDEs respectively:

dIXt=√

2φ⁻¹(Xt)utdBt+φ⁻²(Xt)Z⁰(Xt) dt+N⁰(Xt) dlt, X0=x;

dIYt=√

2φ⁻¹(Yt)P_X⁰_t,YtutdBt+φ⁻²(Yt)Z⁰(Yt) dt+N⁰(Yt) d˜lt, Y0=y.

Then, as explained in the proof of Theorem 2.2, in which we derived (2.9), we have dρ⁰(X_t, Y_t)6√

2(φ⁻¹(X_t)−φ⁻¹(Y_t))h∇⁰ρ⁰(·, Yt)(X_t), u_tdB_ti⁰

−

Z ρ⁰(Xt,Yt) 0

(φ⁻²Ric^Z( ˙γ(s),γ(s)) +˙ Llogφ−2|∇logφ|²)(γ(s)) ds

! dt.

Therefore

dρ⁰(Xt, Yt)²= 2ρ⁰(Xt, Yt) dρ⁰(Xt, Yt) + (φ⁻¹(Xt)−φ⁻¹(Yt))²dt 6d ˜Mt+ 2

Z ρ⁰(X_t,Y_t) 0

h∇⁰φ⁻¹(γ(s)),γ(s)i˙ ⁰ds

!² dt

−2ρ⁰(Xt, Yt)

Z ρ⁰(X_t,Y_t) 0

(φ⁻²Ric^Z( ˙γ(s),γ(s)) +˙ Llogφ−2|∇logφ|²)(γ(s)) dsdt

6 d ˜Mt−2ρ⁰(Xt, Yt)

Z ρ⁰(X_t,Y_t) 0

(φ⁻²(γ(s))Ric^Z( ˙γ(s),γ(s))˙ +Llogφ(γ(s))−3|∇logφ(γ(s))|²) ds

dt 6 d ˜M_t−2K_φ,3ρ⁰(X_t, Y_t)²dt,

where

d ˜Mt= 2√

2ρ⁰(Xt, Yt)(φ⁻¹(Xt)−φ⁻¹(Yt))h∇⁰ρ⁰(·, Yt)(Xt), utdBti⁰. It follows that

W2(δxPt, δyPt)²6E^(x,y)[ρ(Xt, Yt)²]6kφk²_∞E^(x,y)[ρ⁰(Xt, Yt)²] 6kφk²_∞e^−2Kφ,3tρ⁰(x, y)²6kφk²_∞e^−2Kφ,3tρ(x, y)² which completes the proof.

We now investigate Talagrand-type inequalities with respect to the uniform distance on the path space W^T :=C([0, T];M) of the (reflecting) diffusion process, for a given T >0. Let X_t^µ be the (reflecting if

∂M 6=∅) diffusion process generated byLwith initial distributionµ∈P(M). Let Π^T_µ be the distribution of

X_[0,T]^µ :={X_t^µ:t∈[0, T]},

which is a probability measure on the (free) path spaceW^T. When µ =δx we denote Π^T_δ

x = Π^T_x and X_[0,T]^δx =X_[0,T]^x . For any non-negative measurable functionF onW^T such that Π^T_µ(F) = 1, one has

µ^T_F(dx) := Π^T_x(F)µ(dx)∈P(M). (2.18) The the uniform distance onW^T is given by

ρ∞(γ, η) := sup

t∈[0,T]

ρ(γt, ηt), γ, η∈W^T.

LetW₂^ρ∞ be theL²-Wasserstein distance (orL²-transportation cost) induced byρ∞. In general, for any p∈[1,∞) and two probability measures Π1,Π₂ onW^T,

W_p^ρ∞(Π1,Π2) := inf

π∈C(Π₁,Π₂)

Z Z

W^T×W^T

ρ∞(γ, η)^pπ(dγ,dη) 1/p

is theL^p-Wasserstein distance (or L^p-transportation cost) of Π₁ and Π₂, induced by the uniform norm, whereC(Π₁,Π₂) is the set of all couplings for Π₁ and Π₂. Moreover, forF>0 with Π^T_µ(F) = 1, let

µ^T_F(dx) = Π^T_x(F)µ(dx).

The following result improves [15, Theorems 4.1 and 4.2] or [16, Theorems 4.5.3 and 4.5.4]:

Theorem 2.8. If there exists φ∈D and a constant K_φ satisfying

Ric^Z+Llogφ−2|∇logφ|²>K_φ then

(i) forF >0, Π^T_µ(F) = 1 andµ∈P(M), W₂^ρ∞(FΠ^Tµ,Π^T_µT

F)²6 2kφk²∞

Kφ

(e^2Kφ+T−e^2K

− φT

) inf

R>0

n

(1 +R⁻¹) exp 8(1 +R)k∇logφk²∞

o

Π^Tµ(FlogF);

(i’) forF >0, Π^T_µ(F) = 1 andµ∈P(M), W₂^ρ∞(FΠ^Tµ,Π^T_µT

F)²6 2kφk²∞

Kφ

(1−e^−2KφT) inf

R>0

n

(1 +R⁻¹) exp

8(1 +R)k∇logφk²∞e^2Kφ+T o

Π^Tµ(FlogF);

(ii) for any µ, ν∈P(M),

W₂^ρ∞(Π^T_µ,Π^T_ν)62kφk∞e^{(Kφ−+k∇logφk∞)T}W2(µ, ν).

Remark 2.9.

(a) Whenk∇logφk∞>0 andK_φ>0 the upper bound in (i) is better than that in (i’).

(b) When the boundary is convex we can chooseφ≡1. In this case∇logφ= 0 and the estimate in (i’) is consistent with [16, Theorem 4.4.2 (2)] for the convex case.

(c) We note that [16, Theorem 4.4.2 (6)] needs to be corrected as follows:

W₂^ρ∞(Π^T_µ,Π_ν^T)6e^K−TW₂(µ, ν),

where K is the lower bound of Ricci curvature. It is then consistent with Theorem 2.8 (ii) when φ≡1 and the boundary is convex.

Proof of Theorem 2.8.

(i) Simply denote X_[0,T]^x =X_[0,T]. Let F be a positive bounded measurable function on W^T such that infF >0 and Π^T_x(F) = 1. Let

dQ=F(X[0,T]) dP. SinceE

F(X[0,T])

= Π^T_µ(F) = 1,Qis a probability measure on Ω. Then we conclude that there exists a uniqueFt-predictable processβ_t onR^d such that

F(X[0,T]) = exp Z T

0

hβs,dBsi −1 2

Z T 0

kβsk²ds

!

and

Z T 0

EQkβ_sk²ds= 2E

F(X_[0,T]) logF(X_[0,T])

. (2.19)

Then, by the Girsanov theorem, ˜Bt:=Bt−Rt

0βsds,t∈[0, T] is ad-dimensional Brownian motion under the probability measureQ.

As explained in the proof of [16, Theorem 4.5.3], it suffices to assumeµ=δ_x,x∈M. In this case, the desired inequality involves

µ^T_F =δx and Π^T_µ(FlogF) = Π^T_x(FlogF).

Since the diffusion coefficients are non-constant, it is convenient to adopt the Itˆo differential d_I for the Girsanov transformation. So the reflectingLdiffusion processXtcan be constructed by solving the Itˆo SDE

d_IX_t=√

2φ⁻¹(X_t)u_tdB_t+φ⁻²(X_t)Z⁰(X_t) dt+N⁰(X_t) dl_t, X₀=x, whereB_tis thed-dimensional Brownian motion with natural filtrationFt. Then

dIXt=√

2φ⁻¹(Xt)utd ˜Bt+{φ⁻²(Xt)Z⁰(Xt) +√

2φ⁻¹(Xt)utβt}dt+N⁰(Xt) dlt, X0=x (2.20) and letYt solve

d_IY_t=√

2φ⁻¹(Y_t)P_X⁰

t,Y_tu_td ˜B_t+φ⁻²(Y_t)Z⁰(Y_t) dt+N⁰(Y_t) d˜l_t, Y₀=x (2.21) where l_t and ˜l_t are the local times of X_t and Y_t on ∂M, respectively. Moreover, for any bounded measurable functionGonW^T,

EQG(X_[0,T]) :=E(F G)(X_[0,T]) = Π^T_x(F G).

We conclude that the distribution ofX_[0,T] underQcoincides withFΠ^T_x. Therefore, W₂^ρ∞(FΠ^T_x,Π^T_x)²6EQρ_∞(X_[0,T], Y_[0,T])²=EQ max

t∈[0,T]ρ(X_t, Y_t)² 6kφk²_∞EQ max

t∈[0,T]ρ⁰(Xt, Yt)². (2.22)

Note that due to the convexity of the boundary,

hN⁰(x),∇⁰ρ(·, y)(x)i⁰60, x∈∂M.

From this and equations (2.20) and (2.21), it follows that dρ⁰(X_t, Y_t)6√

2(φ⁻¹(X_t)−φ⁻¹(Y_t))D

∇⁰ρ⁰(·, Yt)(X_t), u_td ˜B_tE⁰

−K_φρ⁰(X_t, Y_t) dt+√

2kβtkdt.

Defining

Mt:=√ 2

Z t 0

e^Kφs(φ⁻¹(Xs)−φ⁻¹(Ys))D

∇⁰ρ⁰(·, Ys)(Xs), usd ˜Bs

E0

we have

ρ⁰(X_t, Y_t)6e^−Kφt

M_t+√ 2

Z t 0

e^Kφskβskds

, t∈[0, T].

So to prove (i), we will estimate the function ht=EQ max

s∈[0,t]e^2Kφsρ⁰(Xs, Ys)². By the Doob inequality, for anyR >0, we have

ht: =EQ max

s∈[0,t]e^2Kφsρ⁰(Xs, Ys)² 6(1 +R)EQ max

s∈[0,t]M_s²+ 2(1 +R⁻¹) max

s∈[0,t]EQ

Z s 0

e^Kφrkβrkdr ²

64(1 +R)EQM_t²+ 2(1 +R⁻¹) Z t

0

e^2Kφsds Z t

0

EQkβ_sk²ds 68(1 +R)k∇logφk²_∞

Z t 0

hsds+ 2(1 +R⁻¹) Z T

0

e^2Kφsds Z T

0

EQkβsk²ds, t∈[0, T]. (2.23) Sinceh0= 0, by using the Gronwall inequality, this inequality further implies

h_T 62(1 +R⁻¹) exp 8(1 +R)k∇logφk²_∞ Z T

0

e^2Kφsds Z T

0

EQkβ_sk²ds (2.24) By (2.19) and (2.24) we thus have

EQ max

s∈[0,T]

ρ⁰(X_s, Y_s)²64(1 +R⁻¹) exp 8(1 +R)k∇logφk²_∞e^2K+φT−e^2Kφ−T

2K_φ Π^T_x(FlogF).

(i’) For this we use the function

˜ht= e^2KφtEQ max

s∈[0,t]ρ⁰(Xs, Ys)². The inequality (2.23) should then be modified as follows:

˜ht:= e^2KφtEQ max

s∈[0,t]ρ⁰(Xs, Ys)² 6e^2Kφt(1 +R)EQ max

s∈[0,t]e^−2KφsM_s²+ 2 e^2Kφt(1 +R⁻¹) max

s∈[0,t]EQ

Z s 0

e^{−Kφ(s−r)}kβrkdr 2

64(1 +R) e^2Kφ+tEQM_t²+ 2(1 +R⁻¹) Z t

0

e^2Kφr dr Z t

0

EQkβ_sk²ds 68(1 +R)k∇logφk²_∞e^2Kφ+T

Z t 0

˜hsds+ 2(1 +R⁻¹) Z T

0

e^2Kφr dr Z T

0

EQkβsk²ds, t∈[0, T].

Since ˜h0= 0, this inequality implies h˜_T 62(1 +R⁻¹) exp

8(1 +R)k∇logφk²_∞e^2Kφ+TZ T 0

e^2Kφsds Z T

0

EQkβsk²ds.

We then conclude that EQ max

s∈[0,T]ρ⁰(Xs, Ys)²64(1 +R⁻¹) exp

8(1 +R)k∇logφk²_∞e^2Kφ+T1−e^−2KφT 2Kφ

Π^T_x(FlogF).

(ii) Without loss of generality, we considerµ=δ_x, andν =δ_y. LetX_tandY_tsolve the following SDEs, respectively:

dIXt=√

2φ⁻¹(Xt)utdBt+φ⁻²(Xt)Z⁰(Xt) dt+N⁰(Xt) dlt, X0=x;

dIYt=√

2φ⁻¹(Yt)P_X⁰

t,Y_tutdBt+φ⁻²(Yt)Z⁰(Yt) dt+N⁰(Yt) d˜lt, Y0=y.

Then, as explained in the proof of Theorem 2.2, we have dρ⁰(Xt, Yt)6√

2(φ⁻¹(Xt)−φ⁻¹(Yt))h∇⁰ρ⁰(·, Yt)(Xt), utdBti⁰

−

Z ρ⁰(Xt,Yt) 0

φ⁻²Ric^Z( ˙γ(s),γ(s)) +˙ Llogφ−2|∇logφ|²

(γ(s)) ds

!

dt. (2.25) Therefore,

ρ⁰(Xt, Yt)6e^−Kφt( ˆMt+ρ⁰(x, y)), t>0 (2.26) for

Mˆt:=√ 2

Z t 0

e^Kφs(φ⁻¹(Xs)−φ⁻¹(Ys))h∇⁰ρ(·, Ys)(Xs), usdBsi⁰. Again using the Itˆo formula, we have

dρ⁰(Xt, Yt)²6d ˜Mt−2(Kφ− k∇logφk²_∞)ρ⁰(Xt, Yt)²dt where

d ˜M_t= 2ρ⁰(X_t, Y_t)(φ⁻¹(X_t)−φ⁻¹(Y_t))h∇⁰ρ⁰(·, Yt)(X_t), u_tdB_ti⁰ which implies

Eρ⁰(Xt, Yt)²6e^{−2(Kφ−k∇logφk2∞)t}ρ⁰(x, y)². Combining this with (2.26) we arrive at

W₂^ρ∞(Π^T_x,Π^T_y)²6kφk²_∞E max

t∈[0,T]ρ⁰(Xt, Yt)² 6kφk²_∞e^2K−φTE max

t∈[0,T]( ˆMt+ρ⁰(x, y))² 64kφk²_∞e^2Kφ−TE( ˆMT +ρ⁰(x, y))²

= 4kφk²_∞ e^2K−φT

EMˆ_T²+ρ⁰(x, y)² 64kφk²_∞e^2Kφ−T 2

Z T 0

e^2Kφtk∇logφk²_∞Eρ⁰(X_t, Y_t)²dt+ρ⁰(x, y)²

!

64kφk²_∞e^{2(Kφ−+k∇logφk2∞)T}ρ⁰(x, y)² 64kφk²_∞e^{2(Kφ−+k∇logφk2∞)T}ρ(x, y)²

where the second inequality is due to the Doob inequality. This implies the desired inequality forµ=δx

andν =δy.

Corollary 2.10. If there exists φ∈D and a constantK_φ satisfying

Ric^Z+Llogφ−2|∇logφ|²>Kφ

then

(i) forF >0,Π^T_µ(F) = 1 andµ∈P(M), W₂^ρ∞(FΠ^Tµ,Π^T_µT

F)² 6 2kφk²∞

Kφ

e^2Kφ+T−e^2K

− φT

exp

8k∇logφk²∞+ 4√

2k∇logφk∞

Π^Tµ(FlogF);

(i’) forF >0,Π^T_µ(F) = 1 andµ∈P(M), W₂^ρ∞(FΠ^Tµ,Π^T_µT

F)²6 2kφk²∞

Kφ

1−e^−2KφT exp

8k∇logφk²∞e^2Kφ+T+4√

2k∇logφk∞e^K+φT

Π^Tµ(FlogF).

Proof. It is easily observed that

(1 +R⁻¹) exp 8(1 +R)k∇logφk²_∞

6exp R⁻¹+ 8(1 +R)k∇logφk²_∞ .

Taking the infimum aboutRon the right side above, we arrive at exp R⁻¹+ 8(1 +R)k∇logφk²_∞

>exp

8k∇logφk²_∞+ 4√

2k∇logφk_∞ which allows to prove (i). The inequality (i’) can be checked in the same way.

3 New construction of function log φ

In this section, we give a new construction of a functionφwhich satisfies the conditions of the previous section. To do so, we let ρ∂ be the Riemannian distance to the boundary ∂M and use a comparison theorem for ∆ρ_∂ near the boundary, essentially due to [8]. Note that, by using local charts, it is clear thatρ∂ is smooth in a neighbourhood of∂M. We call

i∂ := sup{r >0 :ρ∂ is smooth on{ρ∂ < r}}

the injectivity radius of∂M. Obviously,i_∂ >0 ifM is compact, but it could be zero in the non-compact case (sup∅= 0 by convention). As [16, Theorem 1.2.3] we have:

Lemma 3.1. Let θ, k be constants such thatII6θ andSect6k. Let

h(t) :=









 cos√

kt−^√θ

ksin√

kt, k >0,

1−θt, k= 0,

cosh√

−kt−^√θ

−ksinh√

−kt, k <0

(3.1)

fort >0. Let h⁻¹(0) be the first zero of h(with h⁻¹(0) :=∞ if h(t)>0 for all t>0). Then for any x∈M˚ such that ρ_∂(x)6i_∂∧h⁻¹(0)we have

∆ρ∂(x)>(d−1)h⁰

h(ρ∂(x)). (3.2)