Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds

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Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds

Marc Arnaudon

, Anton Thalmaier

, Feng-Yu Wang

aSchool of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

bD´epartement de math´ematiques, Universit´e de Poitiers, T´el´eport 2 - BP 30179, F–86962 Futuroscope Chasseneuil Cedex, France

cInstitute of Mathematics, University of Luxembourg, 162A, avenue de la Fa¨ıencerie, L–1511 Luxembourg, Luxembourg dDepartment of Mathematics, Swansea University, Singleton Park, SA2 8PP, Swansea, UK

Received 1 October 2008; received in revised form 1 July 2009; accepted 1 July 2009 Available online 8 July 2009

Abstract

A gradient-entropy inequality is established for elliptic diffusion semigroups on arbitrary complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived.

c

2009 Elsevier B.V. All rights reserved.

MSC:58J65; 58J35; 60H30

Keywords:Harnack inequality; Heat equation; Gradient estimate; Diffusion semigroup

1. The main result

LetM be a non-compact complete connected Riemannian manifold, andP_t be the Dirichlet diffusion semigroup generated byL =∆+ ∇V for someC²functionV. We intend to establish reasonable gradient estimates and Harnack type inequalities forP_t. In case that Ric−Hess_V is bounded below, a dimension-free Harnack inequality was established in [14] which, according

ISupported in part by WIMICS, NNSFC(10721091) and the 973-Project.

∗Corresponding author at: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China. Tel.:

+86 010 5880 8811.

E-mail addresses:wangfy@bnu.edu.cn,F.Y.Wang@swansea.ac.uk(F.-Y. Wang).

0304-4149/$ - see front matter c2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.spa.2009.07.001

to [15], is indeed equivalent to the corresponding curvature condition. See e.g. [2] for equiva- lent statements on heat kernel functional inequalities; see also [8,3,7] for a parabolic Harnack inequality using the dimension–curvature condition by shifting time, which goes back to the classical local parabolic Harnack inequality of Moser [9].

Recently, some sharp gradient estimates have been derived in [11,18] for the Dirichlet semi- group on relatively compact domains. More precisely, forV =0 and a relatively compact open C²domainD, the Dirichlet heat semigroupP_t^Dsatisfies

|∇P_t^Df|(x)≤C(x,t)P_t^Df(x), x∈ D, t>0, (1.1) for some locally bounded functionC:D×]0,∞[→]0,∞[and all f ∈B_b⁺, the space of bounded non-negative measurable functions onM. Obviously, this implies the Harnack inequality

P_t^Df(x)≤ ˜C(x,y,t)P_t^Df(y), t >0, x,y∈ D, f ∈B_b⁺, (1.2) for some functionC˜:M²×]0,∞[→]0,∞[. The purpose of this paper is to establish inequalities analogous to(1.1)and(1.2)globally on the whole manifoldM.

On the other hand however, both(1.1)and(1.2) are, in general, wrong for P_t in place of P_t^D. A simple counter-example is already the standard heat semigroup onR^d. Hence, we turn to search for the following slightly weaker version of gradient estimate:

|∇P_tf(x)| ≤δ

P_t(f log f)−P_t flogP_t f(x)+C(δ,x)

t∧1 P_tf(x),

x∈ M, t>0, δ >0, f ∈B_b⁺, (1.3)

for some positive functionC:]0,∞[×M →]0,∞[. When Ric −Hess_V is bounded below, this kind of gradient estimate follows from [2, Proposition 2.6] but is new without curvature conditions. In particular, it implies the Harnack inequality with power introduced in [14] (see Theorem 1.2).

Theorem 1.1. There exists a continuous positive function F on]0,1] ×M such that

|∇Ptf(x)| ≤δ (Pt flog f −Ptf logPtf) (x) +

F(δ∧1,x) 1 δ(t∧1)+1

+2δ

e

P_t f(x),

δ >0, x∈ M, t >0, f ∈B_b⁺. (1.4)

Theorem 1.2. There exists a positive function C∈C(]1,∞[×M²)such that (P_tf(x))^α ≤(P_t f^α(y))exp

2(α−1)

e +αC(α,x,y) αρ²(x,y)

(α−1)(t∧1)+ρ(x,y) , α >1, t>0, x,y∈ M, f ∈B_b⁺,

whereρ is the Riemannian distance on M. Consequently, for anyδ > 2there exists a positive function C_δ ∈ C([0,∞[×M)such that the transition density p_t(x,y) of P_t with respect to µ(dx):=e^V(x)dx , wheredx is the volume measure, satisfies

p_t(x,y)≤ exp

−ρ(x,y)²/(2δt)+C_δ(t,x)+C_δ(t,y) qµ(B(x,√

2t))µ(B(y,√

2t)) , x,y∈M, t ∈]0,1[.

Remark 1.1. According to the Varadhan asymptotic formula for short time behavior, one has limt→04tlogpt(x,y)= −ρ(x,y)², x6= y. Hence, the above heat kernel upper bound is sharp for short time, asδis allowed to approximate 2.

The paper is organized as follows: In Section2we provide a formula expressingPt in terms of P_t^Dand the joint distribution of(τ,X_τ), whereXt is theL-diffusion process andτ its hitting time to∂D. Some necessary lemmas and technical results are collected. Proposition 2.5 is a refinement of a result in [18] to make the coefficient ofρ(x,y)/tsharp and explicit. In Section3 we use parallel coupling of diffusions together with Girsanov transformation to obtain a gradient estimate for Dirichlet heat semigroup. Finally, complete proofs of Theorems 1.1 and1.2 are presented in Section4.

To prove the indicated theorems, besides stochastic arguments, we make use of a local gradient estimate obtained in [11] forV =0. For the convenience of the reader, we include a brief proof for the case with drift in theAppendix.

2. Some preparations

Let X_s(x)be anL-diffusion process with starting pointxand explosion timeξ(x). For any bounded openC²domainD⊂Msuch thatx ∈ D, letτ(x)be the first hitting time of X_s(x)at the boundary∂D. We have

P_t f(x)=E

f(X_t(x))1_{t<ξ(x)}, P_t^Df(x)=E

f(X_t(x))1_{t<τ(x)}. Letp_t^D(x,y)be the transition density ofP_t^D with respect toµ.

We first provide a formula for the density h_x(t,z) of (τ(x),X_τ(x)(x)) with respect to dt⊗ν(dz), whereνis the measure on∂Dinduced byµ(dy):=e^V(y)dy.

Lemma 2.1. Let K(z,x)be the Poisson kernel in D with respect toν. Then hx(t,z)=

Z

D

−∂tp_t^D(x,y)

K(z,y) µ(dy). (2.1)

Consequently, the density s7→`x(s)of τ(x)satisfies the equation:

`x(s)= Z

D

−∂tp_t^D(x,y)

µ(dy). (2.2)

Proof. Every bounded continuous function f:∂D→Rextends continuously to a functionhon D¯ which is harmonic inDand represented by

h(x)= Z

∂D

K(z,x)f(z) ν(dz).

Recall thatz7→ K(z,x)is the distribution density ofX_τ(x)(x). Hence E[f(X_τ(x)(x))] =h(x)=

Z

∂D

K(z,x)f(z) ν(dz).

On the other hand, the identity h(x)=E[h(X_t∧τ(x)(x))]

yields h(x)=

Z

D

p_t^D(x,y)h(y) µ(dy)+ Z

∂D

ν(dz)Z t 0

h_x(s,z)f(z)ds

= Z

D

p_t^D(x,y)Z

∂D

K(z,y)f(z)ν(dz)

µ(dy)+ Z

∂D

ν(dz)Z t 0

h_x(s,z)f(z)ds

= Z

∂D

f(z)Z

D

p_t^D(x,y)K(z,y) µ(dy)+ Z t

0

hx(s,z)ds ν(dz),

which implies that K(z,x)=

Z

D

p_t^D(x,y)K(z,y) µ(dy)+ Z t

0

h_x(s,z)ds. (2.3)

Differentiating with respect totgives hx(t,z)= −∂t

Z

D

p_t^D(x,y)K(z,y) µ(dy). (2.4)

Since∂tp_t^D(x,y)is bounded on[ε, ε⁻¹] × ¯D× ¯Dfor anyε ∈]0,1[, Eq.(2.1)follows by the dominated convergence.

Finally, Eq.(2.2)is obtained by integrating(2.1)with respect toν(dz). Lemma 2.2. The following formula holds:

P_tf(x)= P_t^Df(x)+ Z

]0,t]×∂D

P_t−sf(z)h_x(s,z)dsν(dz)

= P_t^Df(x)+ Z

]0,t]×∂D

P_t−sf(z)P_s^D_/2h_.(s/2,z)(x)dsν(dz).

Proof. The first formula is standard due to the strong Markov property:

P_tf(x)=E

f(X_t(x))1_{t<ξ(x)}

=E

f(X_t(x))1_{t<τ(x)}

+E

f(X_t(x))1{τ(x)<t<ξ(x)}

= P_t^Df(x)+E E

f(X_t(x))1{τ(x)<t<ξ(x)}|(τ(x),X_τ(x)(x))

= P_t^Df(x)+ Z

]0,t]×∂D

Pt−sf(z)hx(s,z)dsν(dz). (2.5) Next, since

∂sp_s^D(x,y)=L p_s^D(·,y)(x)=L P_s^D_/2p_s^D_/2(·,y)(x)

= P_s/^D₂(L p_s/^D₂(·,y))(x)=P_s/^D₂(∂up_u^D(·,y)|_u=s/2)(x), it follows from(2.1)that

hx(s,z)=P_s^D_/2h_.(s/2,z)(x). (2.6)

This completes the proof.

We remark that formula(2.6)can also be derived from the strong Markov property without invoking Eq.(2.1). Indeed, for anyu <sand any measurable set A ⊂∂D, the strong Markov

property implies that

Pτ(x) >s, X_τ(x)(x)∈ A =E

1_{u<τ(x)}

Pτ(x) >s, X_τ(x)(x)∈ A|Fu

= Z

D

p_u^D(x,y)P

τ(y) >s−u, X_τ(y)(y)∈ A µ(dy), and thus,

hx(s,z)=P_u^Dh_.(s−u,z)(x), s>u >0, x∈ D, z∈∂D.

Lemma 2.3. Let D be a relatively compact open domain andρ∂D be the Riemannian distance to the boundary∂D. Then there exists a constant C>0depending on D such that

P{τ(x)≤t} ≤Ce^−ρ∂²D(x)/16t, x∈ D, t>0.

Proof. Forx ∈ D, let R := ρ∂D(x)andρx the Riemannian distance function to x. Since D is relatively compact, there exists a constantc >0 such that Lρx² ≤ cholds on Doutside the cut-locus ofx. Letγt := ρx(Xt(x)), t ≥ 0. By Itˆo’s formula, according to Kendall [6], there exists a one-dimensional Brownian motionbt such that

dγ_t²≤2

√

2γtdb_t+cdt, t≤τ(x).

Thus, for fixedt >0 andδ >0, Z_s :=exp

δ tγ_s²−δ

tcs−4δ² t²

Z s 0

γ_u²du

, s≤τ(x) is a supermartingale. Therefore,

P{τ(x)≤t} = P

s∈[0,tmax]γs∧τ(x)≥ R

≤P

s∈[0,t]max Zs∧τ(x)≥e^{δR2/t−δc−4δ2R2/t}

≤ exp

cδ−1

t(δR²−4δ²R²) . The proof is completed by takingδ:=1/8.

Lemma 2.4. On a measurable space(E,F,µ)˜ satisfyingµ(E) <˜ ∞, let f ∈ L¹(µ)˜ be non- negative withµ(˜ f) > 0. Then for every measurable functionψ such thatψf ∈ L¹(µ), there˜ holds:

Z

E

ψfdµ˜ ≤ Z

E

f log f

µ(˜ f)dµ˜+ ˜µ(f)log Z

E

e^ψdµ.˜ (2.7)

Proof. This is a direct consequence of [12] Lemma 6.45. We give a proof for completeness.

Multiplying f by a positive constant, we can assume thatµ(˜ f) = 1. IfR

Ee^ψdµ˜ = ∞, then (2.7)is clearly satisfied.

If R

Ee^ψdµ <˜ ∞, then since R

Ee^ψdµ˜ ≥ R

{f>0}e^ψdµ˜, we can assume that f > 0 everywhere. Now from the fact that e^ψ1_f ∈L¹(fµ)˜ , we can apply Jensen’s inequality to obtain

log Z

E

e^ψdµ˜

=log Z

E

e^ψ1 f fdµ˜

≥ Z

E

log

e^ψ1 f

fdµ˜

(note the right-hand-side belongs toR∪ {−∞}). To finish we remark that sinceψf ∈ L¹(µ)˜ , Z

E

log

e^ψ1 f

fdµ˜ =

Z

E

ψf dµ˜ − Z

E

f log fdµ.˜

Finally, in order to obtain precise gradient estimate of the type(1.4), where the constant in front ofρ(x,y)/tis explicit and sharp, we establish the following revision of [18, Theorem 2.1].

Proposition 2.5.Let D be a relatively compact open C²domain in M and K a compact subset of D. For anyε >0, there exists a constant C(ε) >0such that

|∇logp_t^D(·,y)(x)| ≤ C(ε)log(1+t⁻¹)

√ t

+(1+ε)ρ(x,y)

2t ,

t∈]0,1[, x∈K, y∈ D. (2.8)

In addition, if D is convex, the above estimate holds forε=0and some constant C(0) >0.

Proof. Sinceδ:=minKρ∂D >0, it suffices to deal with the case where 0<t ≤1∧δ. To this end, we combine the argument in [18] with relevant results from [16,17]. Lett∈(0,1∧δ],t0= t/2 andy∈ Dbe fixed, and take

f(x,s)=p_s+t^D

0(x,y), x∈ D, s>0.

(a) ApplyingTheorem A.1of theAppendixto the cube

Q:= B(x, ρ_∂D(x))× [s−ρ_∂D(x)²/2,s] ⊂D× [−t₀,t₀], s≤t₀, we obtain

|∇log f(x,s)| ≤ c0

ρ∂D(x)

1+log A f(x,s)

, s≤t₀, (2.9)

whereA :=sup_Q f andc₀>0 is a constant depending on the dimension and curvature onD.

By [7, Theorem 5.2], A≤c1f

x,s+ρ_∂D(x)²

, s∈]0,1], x∈ D, (2.10)

holds for some constantc₁ > 0 depending on DandL. Moreover, by the boundary Harnack inequality of [4] (which treatsZ =0 but generalizes easily to non-zeroC¹driftZ),

f

x,s+ρ_∂D(x)²

≤c₂f(x,s), s∈]0,1], x∈D, (2.11) for some constantc2>0 depending onDandL. Combining(2.9)–(2.11), there exists a constant c>0 depending onDandLsuch that

|∇log f(x,s)| ≤ c

√

s, x∈ D, s∈]0,t₀] withρ_∂D(x)²≤s. (2.12) (b) Let

Ω=n

(x,s):x∈ D, s∈ [0,t₀], ρ_∂D(x)²≥so

andB=sup_Ω f. Since∂sf =L f, for any constantb≥1, we have (L−∂s)

f logb B f

= −|∇f|² f .

Next, again by∂sf =L f and the Bochner–Weizenb¨ock formula, (L−∂s)|∇f|²

f ≥ −2k|∇f|² f ,

wherek≥0 is such that Ric− ∇Z ≥ −konD. Then the function h:= s|∇f|²

(1+2ks)f − f logb B f satisfies

(L−∂s)h ≥0 onD×]0,∞[. (2.13)

Obviouslyh(·,0)≤0, and(2.12)yieldsh(x,s)≤0 fors=ρ_∂D(x)²provided the constantbis large enough. Then the maximum principle and inequality(2.13)implyh≤0 onΩ. Thus,

|∇log f(x,s)|²≤(2k+s⁻¹)logb B

f , (x,s)∈Ω. (2.14)

(c) If Dis convex, by [16, Theorem 2.1] withδ = √

t and t = 2t₀, we obtain (note the generator therein is ¹₂L)

f(x,t₀)= p_2t^D

0(x,y)= p_2t^D

0(y,x)≥c₁ϕ(y)t₀^−d/2e^{−ρ(x,y)2/8t0}, x∈K, y∈ D

for some constantc1 >0, whereϕ >0 is the first Dirichlet eigenfunction of L on D. On the other hand, the intrinsic ultracontractivity forP_t^Dimplies (see e.g. [10])

f(z,s)=p_s+t^D

0(z,y)≤c₂ϕ(y)t₀^−(d+2)/2, z,y∈D, s≤t₀,

for some constantc2>0 depending onD,K andL. Combining these estimates we obtain B

f(x,s) ≤c3t₀⁻¹e^ρ(x,y)2/8t0, x ∈K, s≤t0,

for some constantc₃ > 0 depending onD, K andL. Hence by(2.14)for s = t₀ we get the existence of a constantC >0 such that

|∇logp_2t^D

0(·,y)|²≤(t₀⁻¹+2k)

C+logt₀⁻¹+ρ(x,y)² 8t0

for ally∈ D,x∈ K andt0∈]0,1[witht0≤ρ∂D(x)². This completes the proof by noting that t =2t0.

(d) Finally, ifDis not convex, then there exists a constantσ >0 such that h∇_XN,Xi ≥ −σ|X|², X ∈T∂D,

whereN is the outward unit normal vector field of∂D, andT∂Dis the set of all vector fields tangent to∂D. Letψ ∈ C^∞(D¯)such thatψ = 1 forρ_∂D ≥ ε, 1 ≤ ψ ≤ e^2εσ forρ_∂D ≤ ε, andNlogψ|_∂D ≥σ. By Lemma 2.1 in [17],∂Dis convex under the metricg˜:=ψ⁻²h·,·i. Let

∆˜,∇˜ andρ˜be respectively the Laplacian, the gradient and the Riemannian distance induced by g. By Lemma 2.2 in [17],˜

L:=∆+ ∇V =ψ⁻²h

∆˜ +(d−2)ψ∇ψi + ∇V.

SinceDis convex underg, as explained in the first paragraph in Section 2 of [17],˜ g˜(∇ ˜˜ρ(y,·),∇˜ϕ)|_∂D <0,

so that

σ(y)˜ :=sup

D

g(˜ ∇ ˜˜ρ(y,·),∇˜ϕ) <∞, y∈ D.

Hence, repeating the proof of Theorem 2.1 in [16], but using ρ˜ and ∇˜ in place of ρ and∇ respectively, and taking into account thatψ→1 uniformly asε→0, we obtain

p_2t^D

0(x,y)≥ C₁(ε)ϕ(y)t₀^−d/2e^{−C2(ε)ρ(˜x,y)2/8t0}

≥ C1(ε)ϕ(y)t₀^−d/2e^{−C2(ε)C3(ε)ρ(x,y)2/8t0}

for some constantsC1(ε),C2(ε),C3(ε) >1 withC2(ε),C3(ε)→1 asε→0. Hence the proof is completed.

3. Gradient estimate for Dirichlet heat semigroup using coupling of diffusion processes Proposition 3.1.Let D be a relatively compact C²domain in M. For every compact subset K of D, there exists a constant C=C(K,D) >0such that for allδ >0, t >0, x0 ∈ K and for all bounded positive functions f on M,

|∇P_t^Df(x₀)| ≤δP_t^D

flog f

P_t^Df(x0)

(x₀)+C 1

δ(t∧1)+1

P_t^Df(x₀). (3.1) Proof. We assume thatt∈]0,1[, the other case will be treated at the very end of the proof.

We write∇V = Z so thatL = ∆+Z. Since P_t^D only depends on the Riemannian metric and the vector field Z on the domain D, by modifying the metric andZ outside of Dwe may assume that Ric− ∇Z is bounded below (see e.g. [13]); that is,

Ric− ∇Z ≥ −κ (3.2)

for some constantκ≥0.

Fixx₀∈ K. Let f be a positive bounded function onMandX_s a diffusion with generatorL, starting atx₀. For fixedt≤1, let

v= ∇P_t^Df(x0)

|∇P_t^Df(x₀)|

and denote byu 7→ϕ(u)the geodesics inMsatisfyingϕ(˙ 0)=v. Then d

du u=0

P_t^Df(ϕ(u))=

∇P_t^Df(x0) .

To formulate the coupling used in [1], we introduce some notations.

IfY is a semimartingale inM, we denote by dY its Itˆo differential and by dmY the martingale part of dY: in local coordinates,

dY =

dYⁱ +1

2Γⁱ_{j k}(Y)dhY^j,Y^ki ∂

∂xⁱ

whereΓⁱ_{j k}are the Christoffel symbols of the Levi–Civita connection; if dYⁱ =dMⁱ+dAⁱwhere Mⁱis a local martingale and Aⁱ a finite variation process, then

d_mY =dMⁱ ∂

∂xⁱ.

Alternatively, ifQ(Y):TY₀M →TY_.Mis the parallel translation alongY, then dY_t =Q(Y)td

Z _.

0

Q(Y)⁻¹_s ◦dY_s

t

and

dmYt =Q(Y)tdNt

whereNt is the martingale part of the Stratonovich integralRt

0 Q(Y)⁻¹s ◦dYs. Forx,y∈M, andynot in the cut-locus ofx, let

I(x,y)=

d−1

X

i=1

Z _ρ(x,y)

0

|∇_e˙(x,y)J_i|²+

R(e˙(x,y),J_i)J_i+ ∇_e˙(x,y)Z,e˙(x,y)

s ds (3.3) wheree˙(x,y)is the tangent vector of the unit speed minimal geodesice(x,y)and(J_i)^d_i=1are Jacobi fields alonge(x,y)which together withe˙(x,y)constitute an orthonormal basis of the tangent space atxandy:

Ji(ρ(x,y))=Px,yJi(0), i=1, . . . ,d−1;

herePx,y:TxM →TyMis the parallel translation along the geodesice(x,y).

Letc∈]0,1[. Forh>0 but smaller than the injectivity radius of D, andt >0, letX^hbe the semimartingale satisfyingX₀^h=ϕ(h)and

dX^h_s = P_X

s,X_s^hd_mX_s+Z(X^h_s)ds+ξ_s^hds, (3.4) where

ξ_s^h:=

h ct +κh

n(X_s^h,Xs)

with n(X_s^h,Xs) the derivative at time 0 of the unit speed geodesic from X^h_s to Xs, and P_X

s,Xs^h:T_XsM → T_Xh

sM the parallel transport along the minimal geodesic from X_s toX^h_s. By convention, we putn(x,x)=0 andP_x,x =Id for allx∈ M.

By the second variational formula and(3.2)(cf. [1]), we have dρ(Xs,X_s^h)≤

I(Xs,X_s^h)− h ct −κh

ds≤ −h

ctds, s≤Th,

whereTh :=inf{s ≥ 0 : Xs = X_s^h}. Thus,(Xs,X_s^h)never reaches the cut-locus. In particular, Th≤ctand

X_s =X_s^h, s≥ct. (3.5)

Moreover, we haveρ(X_s,X^h_s)≤hand

|ξ_s^h|²≤h²

κ+ 1 ct

2

. (3.6)

We want to compensate the additional drift ofX^hby a change of probability. To this end, let M_s^h = −

Z s∧ct 0

Dξ_r^h,P_X

r,X^hr d_mX_rE , and

R_s^h=exp

M_s^h−1 2[M^h]_s

.

ClearlyR^his a martingale, and underQ^h=R^h·P, the processX^his a diffusion with generatorL. Lettingτ(x₀)(resp.τ^h) be the hitting time of∂DbyX (resp. byX^h), we have

1_{t<τh}≤1_{t<τ(x0)}+1_{τ(x

0)≤t<τ^h}. But, sinceX^h_s =X_s fors≥ct, we obtain

1_{{τ(x0)≤t<τ}h}=1{τ(x₀)≤ct}1_{t<τh}. Consequently,

1 h

P_t^Df(ϕ(h))−P_t^Df(x₀)

= 1 h E

h

f(X^h_t)R_t^h1{t<τ^h}− f(X_t(0))1{t<τ(x0)}

i

≤ 1 h E

h

f(X^h_t)R^h_t1{t<τ(x0)}− f(X_t(0))1{t<τ(x0)}

i + 1

h E h

f(X^h_t)R_t^h1{τ(x0)≤ct}1{t<τ^h}

i, and sinceX^h_t =Xtthis yields

1 h

P_t^Df(ϕ(h))−P_t^Df(x₀)

≤E

f(X_t)1_{t<τ(x0)}

1

h(R_t^h−1) +1

h E h

f(X^h_t)R_t^h1{τ(x0)≤ct}1_{t<τh}

i. (3.7)

The left hand side converges to the quantity to be evaluated ashgoes to 0. Hence, it is enough to find appropriate lim sup’s for the two terms of the right hand side. We begin with the first term.

Letting Y_s^h=

M_s^h−1 2[M^h]_s

and noting thathn(X_r^h,X_r),P_X

r,X_r^hd_mX_ri =

√

2 db_r up to the coupling timeT_hfor some one- dimensional Brownian motionb_r, we have

R_t^h =exp

M_t^h−1 2[M^h]_t

≤1+M_t^h−1

2[M^h]_t+(Y_t^h)²exp(Y_t^h)

=1+M_t^h− Z t

0

|ξ_s^h|²ds+(Y_t^h)²exp(Y_t^h).

From the assumptions, exp(Y_t^h)andY_t^h/hhave all their moments bounded, uniformly inh >0.

Consequently, since f is bounded, lim sup

h→0

E

f(X_t)1_{t<τ(x0)}

1 h

Z t 0

|ξ_r^h|²dr+(Y_t^h)²exp(Y_t^h)

=0, which implies

lim sup

h→0

E

f(Xt)1{t<τ(x₀)}

1

h(R_t^h−1)

≤lim sup

h→0

E

f(Xt)1{t<τ(x₀)}

1 h

Z s 0

Dξr^h,P_Xr,Xh r dmXr

E . UsingLemma 2.4and estimate(3.6), we have forδ >0

E

f(X_t)1_{t<τ(x0)}

1 h

Z s 0

Dξ_r^h,P_X

r,X_r^hd_mX_rE

≤δP_t^D

f log f

P_t^Df(x0)

(x₀)

+δP_t^Df(x0)logE

1_{t<τ(x0)}exp 1

δh Z ct

0

Dξs^h,P_Xs,Xh

sd_mX_sE

≤δP_t^D

f log f

P_t^Df(x₀)

(x₀)+δP_t^Df(x₀)logE

exp 1

δ²h² Z ct

0

ξ_s^h

2

ds

≤δP_t^D

f log f

P_t^Df(x₀)

(x0)+δP_t^Df(x0)ct δ²

1

c²t²+κ²

≤δP_t^D

f log f

P_t^Df(x0)

(x₀)+ C⁰

cδtP_t^Df(x₀),

whereC⁰ = 1+(cκ)²(recall thatt ≤ 1). Since the last expression is independent ofh, this proves that

lim sup

h→0

E

f(X_t)1_{t<τ(x0)}

1

h(R_t^h−1)

≤δP_t^D

f log f

P_t^Df(x0)

(x0)+ C⁰

cδtP_t^Df(x0). (3.8)

We are now going to estimate lim sup of the second term in Eq.(3.7). By the strong Markov property, we have

E h

f(X^h_t)R_t^h1{τ(x₀)≤ct}1_{t<τh}

i

=EQ^h

h

P_t−ct^D f(X^h_ct)1{τ(x0)≤ct<τ^h}

i

≤ kP_t−ct^D fk_∞Q^h

nτ(x0)≤ct < τ^ho

. (3.9)

Sinceρ(X_s^h,X_s)≤h^ct−s_ct fors∈ [0,ct], we have on{τ(x₀)≤ct< τ^h}: ρ_∂D(X_τ(^h_x

0))≤hct−τ(x0) ct . Fors∈ [0, τ^h−τ(x₀)], define

Y_s⁰=ρ(X_τ(^h_x0)+s, ∂D),

and for fixed smallε > 0 (butε >h), let S⁰ = inf{s ≥ 0, Y_s⁰ = ε or Y_s⁰ =0}. Since under Q^hthe processX^h_s is generated byL, the drift ofρ(X_s^h, ∂D)isLρ(·, ∂D)which is bounded in a neighborhood of∂D. Thus, for a sufficiently smallε >0, there exists aQ^h-Brownian motionβ started at 0, and a constantN >0 such that

Y_s :=h ct−τ(x0)

ct +

√

2βs+N s≥Y_s⁰, s∈ [0,S⁰]. Let

S =inf{u≥0, Y_u=εorY_u=0}. Taking into account that on{τ(x0)=u},

{Y_S⁰0 =ε} ∪ {S⁰>ct−u} ⊂ {Y_S=ε} ∪ {S>ct−u}, we have foru ∈ [0,ct],

Q^h n

ct< τ^h|τ(x₀)=uo

≤Q^h{Y_S⁰ =ε|τ(x₀)=u} +Q^h

S⁰≥ct−u|τ(x₀)=u

≤Q^h{YS=ε|τ(x0)=u} +Q^h{S≥ct−u|τ(x0)=u}

≤Q^h{Y_S=ε|τ(x₀)=u} + 1

ct−uE_Q^h[S|τ(x₀)=u]. Now using the fact that e^{−N Ys} is a martingale andY_s²−2sa submartingale, we get

Q^h{Y_S=ε|τ(x0)=u} = 1−e^{−N hct−uct} 1−e^−Nε ≤C1h and

EQ^h[S|τ(x0)=u] ≤ EQ^h

h

Y_S²|τ(x0)=u i

≤ ε²Q^h{Y_S=ε|τ(x₀)=u}

=ε²1−e^{−N hct−uct}

1−e^−Nε ≤C₂h(ct−u) ct for some constantsC₁,C₂>0. Thus,

Q^h n

ct< τ^h|τ(x0)=u o

≤C1h+ 1 ct−u C2

h(ct−u) ct

≤C1h+C3

h ct ≤C4

h t

for some constantsC3,C4>0 (recall thatt≤1). Denoting by`^hthe density ofτ(x0)underQ^h, this implies

Q^h

nτ(x₀)≤ct< τ^ho

= Z ct

0

`^h(u)Q^h{ct < τ^h|σ^h =u}du

≤ C₄h t

Z ct 0

`^h(u)du

=C₄h

t Q^h{τ(x₀)≤ct}.

In terms of D^−h = {x ∈ D, ρ_∂D(x) > h}andσ^h = inf{s > 0, X^h_s ∈ ∂D^−h}, we have σ^h≤τ(x₀)a.s. Hence, byLemma 2.3,

Q^h{τ(x₀)≤ct} ≤Q^h

nσ^h≤cto

≤Cexp

−ρ_∂D^−h(ϕ(h)) 16ct

,

where we used thatX^h_s is generated byLunderQ^h. This implies Q^h

nτ(x0)≤ct< τ^ho

≤C5

h t exp

−ρ_∂D^−h(ϕ(h)) 16ct

. (3.10)

Since_h¹ P_t^D(ϕ(h))−P_t^D(x0)

converges to|∇P_t^Df(x0)|, we obtain from(3.7)–(3.10),

|∇P_t^Df(x0)| ≤δP_t^D

f log f

P_t^Df(x₀)

(x0)

+ C⁰

cδt P_t^Df(x₀)+C₅kP_t−ct^D fk_∞1 t exp

−ρ_∂D(x0) 16ct

. (3.11)

Finally, as explained in steps (c) and (d) of the proof of Proposition 2.5, for any compact set K ⊂D, there exists a constantC(K,D) >0 such that

kP_t−ct^D fk_∞≤e^C(K,D)/tP_t^Df(x0), c∈ [0,1/2], x0∈ K, t ∈]0,1]. Combining this with(3.11), we arrive at

|∇P_t^Df(x0)| ≤δP_t^D

f log f

P_t^Df(x₀)

(x0)+ C⁰

cδtP_t^Df(x0) +C5

1 t exp

−ρ_∂D(x₀) 16ct

exp

C(K,D) t

P_t^Df(x0). (3.12) Finally, choosingcsuch that

0<c< 1

2 ∧dist(K, ∂D) 16C(K,D), we get for some constantC >0,

|∇P_t^Df(x0)| ≤δP_t^D

f log f

P_t^Df(x₀)

(x0)+C 1

δt +1

P_t^Df(x0),

x₀∈K, δ >0, (3.13)

which implies the desired inequality.

To finish we consider the case t > 1. From the semigroup property, we have P_t^Df = P₁^D(P_t−1^D f). So lettingg=P_t−1^D f and applying(3.13)togat time 1, we obtain

|∇P_t^Df(x₀)| ≤δP₁^D glog g P₁^Dg(x₀)

!!

(x₀)+C 1

δ +1

P₁^Dg(x₀).

Now usingP₁^Dg=P_t^Df, we get

|∇P_t^Df(x0)| ≤δP₁^D(glogg)(x0)−P_t^Df(x0)logP_t^Df(x0)+C 1

δ +1

P_t^Df(x0).

Lettingϕ(x)=xlogx, we have forz∈D glogg(z)=ϕ E

f(Xt−1(z))1{t−1<τ(z)}

≤ E

ϕ f(X_t−1(z))1_{t−1<τ(z)}

=Eϕ(f)(X_t−1(z))1_{t−1<τ(z)}

= P_t−1^D (flog f)(z),

where we successively used the convexity ofϕand the fact thatϕ(0)=0. This implies

|∇P_t^Df(x0)| ≤δP_t^D

f log f

P_t^Df(x0)

(x0)+C 1

δ +1

P_t^Df(x0), which is the desired inequality fort>1.

4. Proof of Theorems1.1and1.2

Proof of Theorem 1.1. We assume that t ∈]0,1[ and refer to the end of the proof of Proposition 3.1for the caset >1. Fixingδ >0 andx0 ∈ M, we take R=160/(δ∧1). LetD be a relatively compact open domain withC²boundary containing B(x0,2R)and contained in B(x0,2R+ε)for some smallε > 0. By the countable compactness of M, it suffices to prove that there exists a constantC = C(D)such that (1.4)holds on B(x0,R)withC in place of F(δ∧1,x₀). We now fixx ∈ B(x₀,R),t ∈]0,1]and f ∈ B_b⁺. Without loss of generality, we may and will assume thatP_tf(x)=1.

(a) LetP_s(x,dy)be the transition kernel of theL-diffusion process, and forx ∈ D,z ∈ M, let

νs(x,dz)= Z

∂D

h_x(s/2,y)P_t−s(y,dz) ν(dy),

whereνis the measure on∂Dinduced byµ(dy)=e^V(y)dy. ByLemma 2.2we have P_tf(x)=P_t^Df(x)+

Z

]0,t]×D×M

p_s^D_/2(x,y)f(z)dsµ(dy)νs(y,dz).

Then

|∇P_tf(x)| ≤ |∇P_t^Df(x)| +

Z

]0,t]×D×M

|∇logp_s^D_/2(·,y)(x)|p_s^D_/2(x,y)f(z)dsµ(dy)νs(y,dz)

=: I1+I2. (4.1)

(b) ByProposition 3.1, we have I₁≤δP_t^D(flog f)(x)+δ

e+C 1

δt +1

, x∈ B(x₀,R), t ∈]0,1[, δ >0 (4.2) for someC=C(D) >0.

(c) ByProposition 2.5withε=1, we have I₂≤

Z

]0,t]×M×D

Clog(√e+s⁻¹)

s +2ρ(x,y) s

p_s^D_/2(x,y)f(z)dsνs(y,dz)µ(dy) (4.3) for some C = C(D) > 0 and all t ∈]0,1]. Applying Lemma 2.4 to the measure µ˜ :=

p_s^D_/2(x,y)dsνs(y,dz)µ(dy)onE :=]0,t] ×M×Dso that µ(E˜ )=P(τ(x)≤t < ξ(x))≤1,

we obtain I2≤ δE

(flog f)(Xt(x))1{τ(x)≤t<ξ(x)} +δ

e+δE

f(Xt(x))1{τ(x)≤t<ξ(x)}

×log Z

]0,t]×M×D

exp

Clog(e+s⁻¹) δ√

s +2ρ(x,y) sδ

ds p_s^D_/2(x,y)νs(y,dz) µ(dy)

≤ δE(f log f)(Xt(x))1{τ(x)≤t<ξ(x)} +δ

e+δE

f(Xt(x))1{τ(x)≤t<ξ(x)}

×log Z

]0,t]×M×D

exp A

δ +9R sδ

ds p_s^D_/2(x,y)νs(y,dz) µ(dy), (4.4)

where

A:=sup

r>0

C

√

rlog(e+r)−r <∞. We get

I₂ ≤ δE

(f logf)(X_t(x))1{τ(x)≤t<ξ(x)}

+δ e +δE

f(Xt(x))1{τ(x)≤t<ξ(x)}

logE

exp(9R/δτ(x)) + A

δ

≤ δE

(f logf)(X_t(x))1{τ(x)≤t<ξ(x)}

+δ

e+δlogE

exp(9R/δτ(x)) +A

≤ δE

(f logf)(X_t(x))1{τ(x)≤t<ξ(x)}

+δlogE

"

exp

9R (δ∧1)τ(x)

^δ∧1_δ #

+A+δ e

=δE(f logf)(X_t(x))1{τ(x)≤t<ξ(x)} +(δ∧1)logE

exp

9R (δ∧1)τ(x)

+A+δ

e. (4.5)

ByLemma 2.3and noting thatρ_∂(x)≥ R, we have E

exp

9R (δ∧1)τ(x)

≤1+E

9R

(δ∧1)τ(x)exp

9R (δ∧1)τ(x)

=1+ Z ∞

0

9Rs (δ∧1)exp

9Rs (δ∧1)

d ds

−P{τ(x)≤s⁻¹}

ds

=1+ 9R (δ∧1)

Z ∞ 0

9R (δ∧1)s+1

exp

9Rs (δ∧1)

P{τ(x)≤s⁻¹}ds

≤1+ 9R (δ∧1)

Z ∞ 0

9R (δ∧1)s+1

exp

9Rs (δ∧1)

exp

−R²s 16

ds

=1+ 9R (δ∧1)

Z ∞ 0

9R (δ∧1)s+1

exp

−Rs (δ∧1)

ds

=1+9 Z ∞

0

(9u+1)exp(−u)du =:A⁰, sinceR=160/(δ∧1). This along with(4.5)yields

I₂≤δE

(f logf)(X_t(x))1{τ(x)≤t<ξ(x)}

+logA⁰+A+δ

e. (4.6)

The proof is completed by combining(4.6)with(4.1),(4.2)and(4.4).

Proof of Theorem 1.2. ByTheorem 1.1,

|∇P_tf(x)| ≤δ (P_t(f log f)(x)−(P_tf)(x)logP_tf(x)) +

F(δ∧1,x) 1

δ(t∧1)+1

+2δ e

P_tf(x), δ >0, x∈M. (4.7)

Forα >1 andx 6=y, letβ(s)=1+s(α−1)and letγ:[0,1] → M be the minimal geodesic fromxtoy. Then| ˙γ| =ρ(x,y). Applying(4.7)withδ=_αρ(x,y)^α−1 , we obtain

d

dslog(Ptf^β(s))^α/β(s)(γs)=α(α−1) β(s)²

P_t(f^β(s)log f^β(s))−(P_tf^β(s))logP_tf^β(s) P_tf^β(s) (γs) + α

β(s)

h∇Ptf^β(s),γ˙si Ptf^β(s) (γs)

≥ αρ(x,y) β(s)P_t f^β(s)(γs)

α−1 αρ(x,y)

Pt(f^β(s)log f^β(s))−(Pt f^β(s))logPt f^β(s) (γs)

− |∇P_tf^β(s)(γs)|

≥ −F

α−1

αρ(x,y)∧1, γs

α²ρ²(x,y)

β(s)(α−1)(t∧1)+αρ(x,y) β(s)

−2(α−1) eβ(s)

≥ −C(α,x,y)

αρ²(x,y)

(α−1)(t∧1)+ρ(x,y)

−2(α−1) e where C(α,x,y) := sup_s∈[0,1]¹_αF

α−1

αρ(x,y)∧1, γs

. This implies the desired Harnack inequality.

Next, for fixedα∈]1,2[, let K(α,t,x)=supn

C(α,x,y):y∈B(x,√ 2t)o

, t >0, x ∈M.

NoteK(α,t,x)is finite and continuous in(α,t,x)∈]1,2[×]0,1[×M. Letp:=2/α. For fixed t∈]0,1[, the Harnack inequality gives fory∈B(x,√

2t), (P_tf(x))²≤(P_tf^α(y))^pexp

2(2−p)

e +2K(α,t,x) 2α α−1 +

√ 2t

. Then, choosingT >t such thatq :=p/2(p−1) <T/t,

µ B(x,√

2t) exp

−2(2−p)

e −2K(α,t,x) 2α α−1+

√ 2t

− t T −qt

(Ptf(x))²

≤ Z

B(x,√

2t)(P_tf^α(y))^pexp

− ρ(x,y)² 2(T−qt)

µ(dy).

Similarly to the proof of [1, Corollary 3], we obtain that for anyδ >2, choosingα= ^2δ

2+δ ∈]1,2[ such thatδ > ₂₋²_α = ^p

p−1 >2, there is a constantc(δ) > 0 such that the following estimate holds:

E_δ(x,t):=

Z

M

pt(x,y)²exp

ρ(x,y)² δt

µ(dy)

≤ expn

c(δ)K(α,t,x)(1+

√ 2t)o µ(B(x,√

2t)) , t >0, x∈ M.

By [5, Eq. (3.4)], this implies the desired heat kernel upper bound forC_δ(t,x):=c(δ)K(α,t,x) (1+

√

2t).

Appendix

The aim of the Appendix is to explain that the arguments in Souplet–Zhang [11] and Zhang [18] for gradient estimates of solutions to heat equations work as well in the case with drift.

Theorem A.1. Let L =∆+Z for a C¹vector field Z . Fix x0∈M and R, T, t0>0such that B(x0,R)⊂M. Assume that

Ric− ∇Z ≥ −K (A.1)

on B(x0,R). There exists a constant c depending only on d, the dimension of the manifold, such that for any positive solution u of

∂tu=Lu (A.2)

on Q_R,T :=B(x0,R)× [t0−T,t0], the estimate

|∇logu| ≤c 1

R +T^−1/2+

√ K



1+log sup

Q_R,T

u u





holds on QR/2,T/2.

Proof. Without loss of generality, let N := sup_QT,Ru = 1; otherwise replaceu byu/N. Let f =loguandω= ^|∇f|2

(1−f)². By(A.2)we have L f + |∇f|²−∂t f =0

so that

∂tω= 2h∇f,∇∂tfi

(1− f)² +2|∇f|²∂tf (1− f)³

= 2h∇f,∇(L f + |∇f|²)i

(1− f)² +2|∇f|²(L f + |∇f|²) (1− f)³

= 2h∇f,∇(∆f + |∇f|²)i

(1− f)² +2|∇f|²(∆f + |∇f|²) (1− f)³ +2h∇_∇fZ,∇fi +2 Hess_f(∇f,Z)

(1− f)² +2|∇f|²hZ,∇fi

(1− f)³ . (A.3)

Moreover,

Lω =∆ω+hZ,∇|f|²i

(1− f)² +2|∇f|²hZ,∇fi (1− f)³

=∆ω+2 Hess_f(∇f,Z)

(1− f)² +2|∇f|²hZ,∇fi

(1− f)³ . (A.4)

Finally, by the proof of [11, (2.9)] with−kreplaced by Ric(∇f,∇f)/|∇f|², we obtain

∆ω−

2h∇f,∇(∆f + |∇f|²)i

(1− f)² +2|∇f|²(∆f + |∇f|²) (1− f)³