Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below

Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below

Marc Arnaudon

, Anton Thalmaier

, Feng-Yu Wang

aDépartement de Mathématiques, Université de Poitiers, Téléport 2 – BP 30179, 86962 Futuroscope Chasseneuil, France

bSchool of Mathematical Sciences, Beijing Normal University, Beijing 100875, China Received 1 September 2005

Available online 21 November 2005

Abstract

Using the coupling by parallel translation, along with Girsanov’s theorem, a new version of a dimension- free Harnack inequality is established for diffusion semigroups on Riemannian manifolds with Ricci curva- ture bounded below by−c(1+ρ_o²), wherec >0 is a constant andρois the Riemannian distance function to a fixed pointoon the manifold. As an application, in the symmetric case, a Li–Yau type heat kernel bound is presented for such semigroups.

2005 Elsevier SAS. All rights reserved.

MSC:58J65; 60H30; 47D07

Keywords:Harnack inequality; Heat kernel; Diffusion semigroup; Curvature

1. Introduction

The difficulties of extending the elliptic Harnack inequality to the parabolic situation are well studied; see the classical work of Moser [19,20], as well as [9,13,14]. In particular, it is in general not possible to compare, for instance on compact sets, different values of a heat semigroupP_tf (for f non-negative) by a constant only depending on t. There are several ways to deal with this deficiency: typically the parabolic Harnack inequality is formulated by introducing a shift

✩ Supported in part by NNSFC (10121101) and RFDP (20040027009).

* Corresponding author.

E-mail addresses:marc.arnaudon@math.univ-poitiers.fr (M. Arnaudon), anton.thalmaier@math.univ-poitiers.fr (A. Thalmaier), wangfy@bnu.edu.cn (F.-Y. Wang).

0007-4497/$ – see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.bulsci.2005.10.001

in time; another possibility is by seeking for Hölder type inequalities with an exponent strictly bigger than 1. For a discussion of the difficulties to obtain Harnack inequalities by probabilistic methods from Bismut type formulas, see for instance [23].

In 1997, a dimension-free Harnack inequality (with an exponent bigger than 1) was estab- lished in [26] for diffusion semigroups with generators having curvature bounded from below.

This inequality has been applied and further developed to the study of functional inequalities (see [1,22,27,29]), heat kernel estimates (see [4,10]), higher order eigenvalues (see [11,28,30]), transportation cost inequalities (see [5]), and short time behavior of transition probabilities (see [2,3,16]). Due to the potential of applications, it would be useful to establish inequalities of this type also for diffusions with curvature unbounded below. On the other hand, since the formula- tion of the inequality in [26] is equivalent to an underlying lower curvature bound (see [31]), the formulation of the resulting inequality will be slightly different in the present paper.

LetM be a connected complete Riemannian manifold of dimension d, either with convex boundary∂M or without boundary. Leto∈Mbe a fixed point,ρ be the Riemannian distance function, andρ_o(x):=ρ(o, x),x∈M. Consider the (reflecting) diffusion semigroupP_t onM generated byL:=+Zfor someC¹-vector fieldZ. We assume that the corresponding (reflect- ing) diffusion process is non-explosive. We shall prove the dimension-free Harnack inequality forP_t under the following condition.

Assumption 1.There exists a constantc >0 such that for allx∈M, Ricx:=inf

Ric(X, X): X∈T_xM, |X| =1 −c

1+ρ_o(x)² , h_Z(x):=sup

∇XZ, X: X∈T_xM, |X| =1

c

1+ρ_o(x) , Z,∇ρ_o(x)c

1+ρ_o(x)

. (1.1)

In this case we have no longer a gradient estimate like |∇P_tf|C_tP_t|∇f|, which has been crucial for deriving the original dimension-free Harnack inequality (cf. the proof of [26, Lemma 2.2]). Under our condition it is possible to prove a weaker type estimate such as

|∇Ptf|^pCtPt|∇f|^p forp >1,

but this is not enough to imply the desired Harnack inequality by following the original proof.

Hence, in this paper we develop a new argument in terms of coupling by parallel translation and Girsanov’s theorem.

The main idea is as follows. Given two pointsx₀=y₀onM, let(x_t, y_t)be the coupling by parallel translation of theL-diffusion process starting from(x₀, y₀). To force the two marginal processes to meet before a given timeT, we make a Girsanov transformation ofy_t, denoted by

˜

y_t, which is equal tox_t att=T and is generated byLunder a weighted probabilityQ:=RP with a densityRinduced by the Girsanov transform. Then, for any bounded measurable function onM, one has

|P_Tf|^α(y₀)=E_Q

f (y˜_T)^α=E

Rf (x_T)^α PT|f|^α(x0)

E[R^α/(α−1)]α−1

, α >1.

To derive a Harnack inequality, it suffices therefore to prove thatE[R^p]<∞forp >1 and to estimate this quantity. We will be able to realize this idea under Assumption 1 (cf. Sections 2 and 3 below for a complete proof).

Theorem 2.Suppose that Assumption1holds. For anyε∈ ]0,1]there exits a constantc(ε) >0 such that

|P_tf|^α(y)P_t|f|^α(x)exp

α(εα+1)ρ(x, y)² 2(2−ε)(α−1)t +c(ε)α²(α+1)²

(α−1)³

1+ρ(x, y)²

ρ(x, y)²+α−1 2

1+ρ_o(x)²

holds for all α >1,t >0, x, y∈M and any bounded measurable functionf on M, where ρ(x, y)is the Riemannian distance fromxtoyandρ_o(x)=ρ(o, x).

As an application of the above Harnack inequality, we present a heat kernel estimate as in [10]. Assume thatZ= ∇V for someC²-functionV onM, such thatP_t is symmetric w.r.t. the measureµ(dx):=e^{V (x)}dx, where dx is the Riemannian volume measure. Letp_t(x, y)be the transition density ofP_t w.r.t.µ; that is,

Ptf (x)=

M

pt(x, y)f (y)µ(dy), x∈M, t >0, f∈Cb(M).

Corollary 3.Suppose that Assumption1 holds and letZ= ∇V. For anyδ >2there exists a constantc(δ) >0such that for anyt >0,

p_t(x, y)exp

−^ρ(x,y)_2δt ² +c(δ)(1+t+t²+ρ_o(x)²+ρ_o(y)²)

µ(B(x,√

2t ))µ(B(y,√ 2t))

, x, y∈M,

whereB(x, r)is the geodesic ball centered atxinMwith radiusr.

2. Proof of Theorem 2 without cut-locus

To explain our argument in a simple way, we assume in this section that the cut-locus is empty;

that is, Cut(M):= {(x, y)∈M×M: x∈cut(y)} = ∅. In the next section, we then treat the technical details for Cut(M)= ∅. Moreover, if∂Mis convex, we may assume thatMis a regular domain in a Riemannian manifold such that the minimal geodesic linking any two points inM is contained inM, see [32, Proposition 2.1.5]. Thus, according to the proof of [25, Lemma 2.1], the reflection of the two marginal processes at the boundary makes them move together faster.

Hence, without loss of generality, we may and will assume that∂M= ∅. Finally, in the sequel we assume thatf is a non-negative measurable bounded function onM.

We now recall the construction of coupling by parallel translation. LetB_t be ad-dimensional Brownian motion. Then theL-diffusion process starting atx₀∈Mcan be constructed by solving the following SDE:

dxt=√

2Φ_t◦dBt+Z(x_t)dt, x₀∈M, (2.1)

whereΦ_t denotes the horizontal lift ofx_t; that is dΦt=H_Φt ◦dxt, Φ₀∈Ox0(M),

in terms of the horizontal lift operatorH:π^∗T M→TO(M).

For given pointsx=y, lete(x, y):[0, ρ(x, y)] →Mbe the unique minimal geodesic fromx toyand letP_x,y:T_xM→T_yMbe the parallel translation along the geodesice(x, y). In particu- larP_x,x=I, the identity operator. Consider the Itô equation

d^Itôy_t=√

2P_xt,ytΦ_tdBt+Z(y_t)dt, y₀∈M, (2.2)

where in coordinates the Itô differential is given by (d^Itôy_t)^k=dy_t^k+1

2

i,j

Γ_ij^k(y_t)d[y_tⁱ, y^j_t],

see Emery [8]. Recall that (2.2) is equivalent to the system of equations dΨt=H_Ψt◦dyt, Ψ₀∈Oy0(M),

dyt=√

2Ψ_t◦dB_t+Z(y_t)dt, y₀∈M, dB_t=Ψ_t⁻¹Px_t,y_tΦtdBt,

where the last equation is an Itô equation inR^dandΨ_t is the horizontal lift ofy_t. See [17, (2.1)]

for an analogous construction (with the mirror reflection operator). SinceP_x,yis smooth,y_t is a well-definedL-diffusion process starting aty₀. We call the pair(x_t, y_t)the coupling by parallel translation of theL-diffusion process.

To calculate the distance processρ(x_t, y_t), letM_x,y:T_xM→T_yM be the mirror reflection operator along the geodesice(x, y); that is,M_x,yX:=P_x,yXifX⊥ ˙e, whileM_x,yX:= −P_x,yX ifX ˙e at the point x. Let{uⁱ}^d_i=⁻₀¹ be an orthonormal basis inR^d such thatΦ_tu⁰= ˙eatx_t. Definevⁱ:=(Ψ_t⁻¹P_xt,ytΦ_t)uⁱ,i=0, . . . , d−1. SinceΦ_tuⁱ,e˙(x_t)=0 for alli=0, we have

v⁰= −(ΨtMx_t,y_tΦt)u⁰, vⁱ=(ΨtMx_t,y_tΦt)uⁱ, i=0.

Then [17, Theorem 2 and (2.5)] implies

dρ(xt, y_t)I_Z(x_t, y_t)dt, tτ, (2.3)

whereτ:=inf{t0:xt=yt}is the coupling time and

IZ(x, y)=

d−1

i=1 ρ(x,y)

0

|∇e(x,y)˙ Ji|²− R

˙

e(x, y), Ji

e(x, y), J˙ i

sds +Zρ(·, y)(x)+Zρ(x,·)(y).

HereRdenotes the Riemann curvature tensor,e(x, y)˙ the tangent vector of the geodesice(x, y), and{Ji}^d_i=⁻₁¹are Jacobi fields alonge(x, y)which, together withe(x, y), constitute an orthonor-˙ mal basis of the tangent space atxandy:

J_i

ρ(x, y)

=P (x, y) J_i(0), i=1, . . . , d−1.

To calculateIZ(xt, yt)we may take(Φt(uⁱ))atxt and(Ψt(vⁱ))atyt. Let K(x, y):= sup

z∈e(x,y)

(−Ricz)⁺, δ(x, y):=sup

∇XZ, Xz: z∈e(x, y), X∈T_zM, |X| =1 .

We have

Zρ(·, y)(x)+Zρ(x,·)(y)=

ρ(x,y)

0

∇e(x,y)_˙ Z,e(x, y)˙

sdsδ(x, y)ρ(x, y).

Thus, by [32, Theorem 2.1.4] (see also [7] and [6]), we obtain IZ(x, y)2

K(x, y)(d−1)tanh

ρ(x, y) 2

K(x, y)/(d−1)

+δ(x, y)ρ(x, y). (2.4) To construct a coupling such that the coupling time is less than a givenT >0, let us consider the equation

d^Itôy˜t=√

2P_xt,y˜tΦtdBt+Z(y˜t)dt

−

I_Z(x_t,y˜_t)+ρ(x₀, y₀) T

n(y˜_t, x_t)dt, y˜₀=y₀, (2.5) wheren(y, x):= ˙e(y, x)|y= ∇ρ(x,·)(y)∈T_yMforx=y. Sincen(x, y)is smooth outside the diagonalD:= {(x, x): x ∈M}, the solutiony˜_t exists and is unique up to the coupling time

˜

τ:=inf{t0:x_t= ˜y_t}. We lety˜_t=x_t fortτ˜. As in (2.3) we have dρ(xt,y˜t)−ρ(x₀, y₀)

T dt, tτ ,˜ so thatτ˜T. Let

N_t:= 1

√2

t∧ ˜τ

0

P_xs,y˜sΦ_sdBs,

I_Z(x_s,y˜_s)+ρ(x0, y0) T

n(y˜_s, x_s)

,

R_t:=exp

N_t−1 2[N]t

. (2.6)

By Girsanov’s theorem, { ˜yt}is an L-diffusion under the weighted probability measure Q:=

R_TP. Therefore, PTf (y)=EQ

f (y˜T)

=E

RTf (xT)

E

f^α(xT)1/α

(ER^β_T)^1/β, α⁻¹+β⁻¹=1. (2.7) By (2.4) and (2.6) we have

[N]T 1 2

T

0

I_Z(x_t,y˜_t)+ρ(x₀, y₀) T

2

dt

1

2 T 0

2

(d−1)K(xt,y˜t)+δ(xt,y˜t)ρ(xt,y˜t)+ρ(x₀, y₀) T

2

dt.

Exploiting the conditions (1.1) and the fact thatρ(x_t,y˜_t)ρ(x₀, y₀), we obtain, givenε∈ ]0,1], [N]T

T

0

c₁

1+ρ(x₀, y₀)²

1+ρ_o(x_t)²

+ρ(x₀, y₀)²

(2−ε)T² dt (2.8)

for some constantc₁=c₁(ε)1. Next, by (1.1) and the Laplacian comparison theorem, we get dρo(x_t)√

2 dbt

+ c

1+ρ_o(x_t)²

/(d−1)coth

ρ_o(x_t)

c

1+ρ_o(x_t)²

/(d−1)

dt +c

1+ρ_o(x_t) dt=:√

2 dbt+U_tdt,

whereb_tis a one-dimensional Brownian motion. In particular, this gives dρo(x_t)²2√

2ρ_o(x_t)dbt+

2Utρ_o(x_t)+2 dt

2√

2ρ_o(x_t)dbt+c₂

1+ρ_o(x_t)² dt for some constantc₂>0. Thus, for anyγ , δ >0, we have

de^{γ (1+ρo(xt)2)e−δt} dMt−γe^−δt

(δ−c₂)

1+ρ_o(x_t)²

−4γe^−δtρ_o(x_t)²

e^{γ (1+ρo(xt)2)e−δt}dt for some local martingaleM_t. Lettingδ:=c₂+4γ we arrive at

E exp

γ

1+ρ_o(x_t)² e^−δt

exp

γ

1+ρ_o(x₀)² . Therefore, there exists a constantδ₀∈ ]0,1]such that

E exp

δ₀

1+ρ_o(x_t)²

exp

1+ρ_o(x₀)²

, t1. (2.9)

Letp:=1+ε,q:=(1+ε)/ε. The proof of Theorem 2 is completed in two more steps.

I. Assume thatT T₀:=2δ0/[c₁β(βp−1)q(1+ρ(x₀, y₀)²)]. We have γ:=β(βp−1)qc1

1+ρ(x₀, y₀)²

T /2δ₀. Then, by (2.8) and (2.9), we obtain

E exp

β(βp−1)q[N]T/2

exp

β(βp−1)qρ(x0, y₀)²

2(2−ε)T E

exp

γ T

T 0

1+ρo(xt)² dt

exp

β(βp−1)qρ(x0, y₀)² 2(2−ε)T

1 T

T

0

E exp

δ₀

1+ρ_o(x_t)² dt

exp

β(βp−1)qρ(x0, y₀)²

2(2−ε)T +1+ρ_o(x₀)² . Combining this with (2.7) and the fact that

E[R_T^β] =E exp

βNT −pβ²[N]T/2 exp

β(βp−1)[N]T/2

Eexp

pβN_T −p²β²[N]T/21/p Eexp

β(βp−1)q[N]T/21/q

= Eexp

β(βp−1)q[N]T/21/q

, we conclude that

(PTf )^α(y0)PTf^α(x0)exp

α(pβ−1)ρ(x0, y₀)² 2(2−ε)T ∇ + α

qβ

1+ρo(x0)² . (2.10)

II. IfT > T₀, then by (2.10) and Jensen’s inequality, (P_Tf )^α(y₀)

P_T0(P_T−T0f )α

(y₀) P_T0(P_T−T0f )^α(x₀)exp

α(pβ−1)ρ(x0, y0)² 2(2−ε)T0 + α

qβ

1+ρ_o(x₀)²

P_Tf^α(x₀)exp

c₂αβ(2β−1)²

1+ρ(x₀, y₀)²

ρ(x₀, y₀)² + α

2β

1+ρ_o(x₀)²

for somec₂=c₂(ε). In conclusion, combining this with (2.10), the desired inequality for some c(ε) >0 is obtained.

3. Proof of Theorem 2 with cut-locus

As already explained in Section 2, we may assume that∂M= ∅. When Cut(M)= ∅, the idea (originally due to Cranston [7]) is to construct the coupling by parallel translation outside of Cut(M)and to let the two marginal processes move independently on the cut-locus. This idea has been realized in [32] by an approximation argument. Since the present situation is different due to the Girsanov transformation, we reformulate the procedure here in detail for our purpose of achieving a coupling time smaller than a givenT >0.

By Itô’s formula it is easy to see that, when Cut(M)= ∅, the generator of the coupling(x_t,y˜_t) is (cf. the proof of [15, Theorem 6.5.1])

L(x)+L(y)+2 d i,j=1

Px,yXi(x), Yj(y)

Xi(x)Yj(y)−

IZ(x, y)+ρ(x0, y0)/T n(y, x), whereL(x)andL(y)denote the operatorLacting on the first and the second components re- spectively, and{Xi}and{Yi}are local frames normal atx andy, respectively. Note that this operator is independent of the choices of the local frames. Thus, in the general situation, we intend to construct a process generated by

L(x, y)˜ :=L(x)+L(y)+21_Cut(M)(x, y) d i,j=1

P_x,yX_i(x), Y_j(y)

X_i(x)Y_j(y)

−

(1_Cut(M)I_Z)(x, y)+ρ(x₀, y₀)/T

n(y, x), (3.1)

wheren(y, x)is set to be zero on Cut(M)∪D. To this end, we adopt an approximation argument as in [32, §2.1].

For anyn1 andε∈ ]0,1[, leth_n,ε∈C^∞(M×M)such that

0h_n,ε1−ε, h_n,ε|Cut(M)n=1−ε, h_n,ε|Cut(M)2n=0, where

Cut(M)n:=

(x, y): ρ_M×M

(x, y),Cut(M)

1/n

, n1, andρ_M×M is the Riemannian distance onM×M. Consider the operator

L˜_n,ε(x, y):=L(x)+L(y)+2hn,ε(x, y) d i,j=1

P_x,yX_i(x), Y_j(y)

X_i(x)Y_j(y)

−

I_Z(x, y)+

1−h_n,ε(x, y)

J (x, y)+ρ(x₀, y₀)/T n(y, x), withJ∈C^∞((M×M)\D)such that

Lρ(·, y)(x)+Lρ(x,·)(y)J (x, y), x=y.

SinceL˜_n,ε is a uniformly elliptic second order differential operator with smooth diffusion coef- ficients and the drift is smooth outside ofD, it generates a unique diffusion process(x_t, y_t^n,ε) up to the coupling timeτ_n,ε:=inf{t0: x_t =y_t^n,ε}(cf. [24, Theorem 6.4.3]), which can be constructed by solving (2.1) and the Itô SDE

d^Itôy_t^n,ε=

2hn,ε(x_t, y_t^n,ε) P_x

t,y_t^n,εΦ_tdBt

+ 2

1−h_n,ε(x_t, y_t^n,ε)

Ψ_t^n,εdB_t+Z(y_t^n,ε)dt

−

I_Z(x_t, y_t^n,ε)+

1−h_n,ε(x_t, y_t^n,ε)

J (x_t, y_t^n,ε)+ρ(x₀, y₀)/T

n(y_t^n,ε, x_t)dt with initial conditiony^n,ε₀ =y0, where Φt andΨ_t^n,ε are the horizontal lifts of xt andy_t^n,ε re- spectively, andBtandB_tare two independentd-dimensional Brownian motions. Indeed, the last equation may be solved first neglecting the drift term involvingn(y_t^n,ε, x_t)and then by applying Girsanov’s theorem. We sety_t^n,ε=x_t fortτ_n,ε. By the choice ofJ and using Itô’s formula of the radial process as presented in [18], we get

dρ(xt, y_t^n,ε)2

1−h_n,ε(x_t, y_t^n,ε)db^n,ε_t −ρ(x₀, y₀)

T dt, tτ_n,ε, (3.2)

whereb_t^n,ε is a one-dimensional Brownian motion. Therefore, lettingP^xn,ε^0,y0 be the distribution of(x_t, y_t^n,ε)_t∈[0,T], where here and in the sequel,(ξ_., η_.)∈C([0, T];M×M)is the canonical path, we have

lim sup

N→∞ sup

n,εP^xn,ε^0,y0

sup

t∈[0,T]ρ(ξ_t, η_t)N

=0. (3.3)

Since the first marginal distribution ofP^xn,ε^0,y0 isP^x0, the distribution of theL-diffusion process starting atx0, it follows from (3.3) that

P^xn,ε^0,y0

sup

s,t∈[0,T]ρM×M

(ξs, ηs), (ξt, ηt)

N

P^xn,ε^0,y0

sup

s,t∈[0,T]

2ρ(ξs, ξ_t)+ρ(ξ_s, η_s)+ρ(ξ_t, η_t)

N

P^xn,ε^0,y0

sup

s,t∈[0,T]ρ(ξs, ξt)N/4

+P^xn,ε^0,y0

sup

t∈[0,T]ρ(ξt, ηt)N/4

=P^x0 sup

s,t∈[0,T]ρ(ξ_s, ξ_t)N/4

+P^xn,ε^0,y0

sup

t∈[0,T]ρ(ξ_t, η_t)N/4

→0,

as N→ ∞. Thus, by [21, Lemma 4] the family{P^xn,ε^0,y0: n1, ε∈ ]0,1[} is tight. We take n_k→ ∞andε→0 such thatP^xnk^0,y,ε⁰ converges weakly to someP^xε^0,y0 (1) ask→ ∞while

P^xε^0,y0 converges weakly to someP^x0,y0 as → ∞. It is trivial to see that P^x0,y0 solves the martingale problem forL˜ up to the coupling time; that is, for anyf ∈C₀^∞((M×M)\D),

f (ξ_t, η_t)− t

0

Lf (ξ˜ _s, η_s)ds, tT ,

is aP^x0,y0-martingale w.r.t. the natural filtration up to inf{t0: ξ_t=η_t}. Moreover, according to the proof of [32, Theorem 2.1.1] and (3.2), there holds

dρ(ξt, ηt)−ρ(x₀, y₀)

T dt, P^x0,y0-a.s. (3.4)

Hence, there exist two independentd-dimensional Brownian motionsBt andB_t on a complete probability space(Ω,F,Ft,P), and two processesxt andy˜t onMsuch that Eq. (2.1) and

d^Itôy˜_t=√

2 1_Cut(M)(x_t,y˜_t)P_xt,ytΦ_tdBt+√

2 1Cut(M)(x_t,y˜_t)Ψ_tdB_t+Z(y˜_t)dt

−

IZ(xt,y˜t)+ρ(x0, y0)/T

n(y˜t, xt)dt, tτ ,˜

hold, whereΦ_t andΨ_t are the horizontal lifts ofx_t andy˜_t respectively, andτ˜:=inf{t0: x_t=

˜

yt}. Moreover, by (3.4) we have dρ(xt,y˜t)−ρ(x₀, y₀)

T dt,

as well asτ˜T. Lety_t=x_t fortτ˜and letR_t be defined as in Section 2 with N_t= 1

√2

t∧ ˜τ

0

P_xs,y˜s

1_Cut(M)(x_s,y˜_s)Φ_sdBs+1Cut(M)(x_s,y˜_s)Ψ_sdB_s ,

IZ(xs,y˜s)+ρ(x₀, y₀) T

n(y˜s, xs)

.

We conclude thaty˜t is generated byL under the probabilityQ:=RTP. The remainder of the proof is analogous to the case where Cut(M)= ∅.

4. Proof of Corollary 3

Givent >0, letT >0,p∈ ]1,2[andq:=p/2(p−1)be such thatqt < T. Applying Theo- rem 2 withα:=2/pandε=1, we obtain, for any bounded non-negative measurable functionf,

I:=µ B(x,√

2t )

e^{−c1(1+t+t2+ρo(x)2)−t /(T−qt )}

P_tf (x)2

B(x,√ 2t )

P_tf^α(y)p

exp

−c₁

1+t+t²+ρ_o(x)²

− t T −qt +α(α+1)p

α−1 +2p c(1)α²(α+1)²(1+2t )t (α−1)³ +p

2(α−1)

1+ρo(y)² µ(dy).

Since onB(x,√

2t)one hasρo(y)²2ρo(x)²+4t, there exists a constantc1=c1(p) >0 such that

I

B(x,√ 2t )

P_tf^α(y)p

exp

−1 2

ρ(x, y)²

(T −qt ) µ(dy).

Combining this with [10, (2.9)] we arrive at I

M

f²(y)exp

−ρ(x, y)²

2T µ(dy). (4.1)

Takingf (y):=(n∧p_t(x, y))exp{n∧ ^ρ(x,y)_2T ²},y∈M, we obtain from (4.1) that

M

n∧p_t(x, y)2

exp

n∧ρ(x, y)² 2T µ(dy) exp{c1(p)(1+t+t²+ρ_o(x)²)+t /(T −qt )}

µ(B(x,√

2t ) .

For anyδ >2, lettingT :=δt /2 andq:=1/2+δ/4, we obtain E_δ(x, t ):=

M

p_t(x, y)²exp

ρ(x, y)²

δt µ(dy)exp{c(δ)(1+t+t²+ρo(x)²)} µ(B(x,√

2t) for somec(δ) >0. Therefore, by [12, (3.4)] we have

pt(x, y)exp

−ρ(x, y)²

2δt Eδ(x, t )Eδ(y, t )

exp{c(δ)(1+t+t²+ρ_o(x)²+ρ_o(y)²)−ρ(x, y)²/(2δt )} µ(B(x,√

2t)µ(B(y,√ 2t))

.

References

[1] S. Aida, Uniform positivity improving property, Sobolev inequalities, and spectral gaps, J. Funct. Anal. 158 (1) (1998) 152–185.

[2] S. Aida, H. Kawabi, Short time asymptotics of a certain infinite dimensional diffusion process, in: Stochastic Analy- sis and Related Topics, VII, Kusadasi, 1998, in: Progr. Probab., vol. 48, Birkhäuser Boston, Boston, MA, 2001, pp. 77–124.

[3] S. Aida, T. Zhang, On the small time asymptotics of diffusion processes on path groups, Potential Anal. 16 (1) (2002) 67–78.

[4] D. Bakry, M. Ledoux, Z. Qian, Logarithmic Sobolev inequalities, Poincaré inequalities and heat kernel bounds, unpublished manuscript, 1997.

[5] S.G. Bobkov, I. Gentil, M. Ledoux, Hypercontractivity of Hamilton–Jacobi equations, J. Math. Pures Appl.

(9) 80 (7) (2001) 669–696.

[6] M.-F. Chen, F.-Y. Wang, Application of coupling method to the first eigenvalue on manifold, Sci. China Ser. A 37 (1) (1994) 1–14.

[7] M. Cranston, Gradient estimates on manifolds using coupling, J. Funct. Anal. 99 (1) (1991) 110–124.

[8] M. Emery, Stochastic Calculus in Manifolds, Springer-Verlag, Berlin, 1989, with an appendix by P.-A. Meyer.

[9] E.B. Fabes, D.W. Stroock, A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash, Arch.

Rational Mech. Anal. 96 (4) (1986) 327–338.

[10] F.-Z. Gong, F.-Y. Wang, Heat kernel estimates with application to compactness of manifolds, Q. J. Math. 52 (2) (2001) 171–180.

[11] F.-Z. Gong, F.-Y. Wang, On Gromov’s theorem andL²-Hodge decomposition, Int. J. Math. Math. Sci. 1–4 (2004) 25–44.

[12] A. Grigor’yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geom. 45 (1) (1997) 33–52.

[13] A. Grigor’yan, L. Saloff-Coste, Stability results for Harnack inequalities, Ann. Inst. Fourier (Grenoble) 55 (3) (2005) 825–890.

[14] W. Hebisch, L. Saloff-Coste, On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier (Grenoble) 51 (5) (2001) 1437–1481.

[15] E.P. Hsu, Stochastic Analysis on Manifolds, Graduate Studies in Mathematics, vol. 38, American Mathematical Society, Providence, RI, 2002.

[16] H. Kawabi, The parabolic Harnack inequality for the time dependent Ginzburg–Landau type SPDE and its applica- tion, Potential Anal. 22 (1) (2005) 61–84.

[17] W.S. Kendall, Nonnegative Ricci curvature and the Brownian coupling property, Stochastics 19 (1–2) (1986) 111–

129.

[18] W.S. Kendall, The radial part of Brownian motion on a manifold: a semimartingale property, Ann. Probab. 15 (4) (1987) 1491–1500.

[19] J. Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961) 577–591.

[20] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964) 101–134, Correction: Comm. Pure Appl. Math. 20 (1967) 231–236.

[21] M.-K. Renesse, Intrinsic coupling on Riemannian manifolds and polyhedra, Electronic J. Probab. 9 (2004) 411–435.

[22] M. Röckner, F.-Y. Wang, Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds, Forum Math. 15 (6) (2003) 893–921.

[23] D.W. Stroock, An Introduction to the Analysis of Paths on a Riemannian Manifold, Mathematical Surveys and Monographs, vol. 74, American Mathematical Society, Providence, RI, 2000.

[24] D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wis- senschaften, vol. 233, Springer-Verlag, Berlin, 1979.

[25] F.-Y. Wang, Application of coupling methods to the Neumann eigenvalue problem, Probab. Theory Related Fields 98 (3) (1994) 299–306.

[26] F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (3) (1997) 417–424.

[27] F.-Y. Wang, Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants, Ann.

Probab. 27 (2) (1999) 653–663.

[28] F.-Y. Wang, Functional inequalities, semigroup properties and spectrum estimates, Infin. Dimen. Anal. Quantum Probab. Relat. Top. 3 (2) (2000) 263–295.

[29] F.-Y. Wang, Logarithmic Sobolev inequalities: conditions and counterexamples, J. Operator Theory 46 (1) (2001) 183–197.

[30] F.-Y. Wang, Functional inequalities and spectrum estimates: the infinite measure case, J. Funct. Anal. 194 (2) (2002) 288–310.

[31] F.-Y. Wang, Equivalence of dimension-free Harnack inequality and curvature condition, Integral Equations Operator Theory 48 (4) (2004) 547–552.

[32] F.-Y. Wang, Functional Inequalities, Markov Properties, and Spectral Theory, Science Press, Beijing/New York, 2004.