# Hamilton’s Harnack inequality and the $W$-entropy formula on complete
Riemannian manifolds

Xiang-Dong Li

Academy of Mathematics and Systems Science, Chinese Academy of Sciences
55, Zhongguancun East Road, Beijing, 100190, P. R. China
E-mail<EMAIL_ADDRESS>Research supported by NSFC No. 10971032, Key Laboratory
RCSDS, CAS, No. 2008DP173182, and a Hundred Talents Project of AMSS, CAS.

Abstract. In this paper, we prove Hamilton’s Harnack inequality and Hamilton
type gradient estimates of the logarithmic heat kernel for the Witten
Laplacian on complete Riemainnian manifolds. As application, we prove the
$W$-entropy formula for the Witten Laplacian on complete Riemannian manifolds
with natural Bakry-Emery Ricci curvature condition. We also prove a family of
logarithmic Sobolev inequalities on complete Riemannian manifolds with
$\mu$-bounded geometry condition.

##  1 Introduction

In a seminal paper [41], Perelman proved the $W$-entropy formula for the Ricci
flow on compact manifolds. As a consequence, the $W$-entropy of the Ricci flow
is nondecreasing in time and the monotonicity is strict except that $M$ is a
shrinking Ricci solitons, i.e., for some $f\in C^{\infty}(M)$ and $\tau>0$, it
holds

$\displaystyle Ric+\nabla^{2}f={g\over 2\tau}.$

As was pointed out in [41], this result “removes the major stumbling block in
Hamilton’s approach to geometrization”. See also [8, 10, 21, 35].

During recent years, many people have established the $W$-entropy formula for
various geometric evolution equations [9, 28, 29, 39, 40, 14, 31, 23]. In [39,
40], Ni proved an analogue of Perelman’s $W$-entropy formula for the linear
heat equation associated with the Laplace-Beltrami operator on complete
Riemannian manifolds with fixed Riemannian metrics. More precisely, let $M$ be
an $n$-dimensional complete Riemannian manifold with a fixed Riemannian metric
$g$, and let $u(x,\tau)={e^{-f(x,\tau)}\over(4\pi\tau)^{n/2}}$ be the heat
kernel to the heat equation

$\displaystyle\partial_{\tau}u=\Delta u.$

The $W$-entropy for the above heat equation is defined by

$\displaystyle W(u,\tau)=\int_{M}\left[\tau|\nabla
f|^{2}+f-n\right]{e^{-f}\over(4\pi\tau)^{n/2}}dv.$

Under some technical assumptions, Ni [39, 40] proved the following entropy
formula

$\displaystyle{dW(u,\tau)\over d\tau}$ $\displaystyle=$
$\displaystyle-2\int_{M}\tau\left(\left|\nabla^{2}f-{g\over
2\tau}\right|^{2}+Ric(\nabla f,\nabla f)\right)udv,$

which implies that $W(u,\tau)$ is decreasing in $\tau$ on complete Riemannian
manifolds with non-negative Ricci curvature. See also [11].

In our previous paper [28], the author further extended the $W$-entropy
formula to the heat equation associated with the Witten Laplacian on complete
Riemannian manifolds. More precisely, let $M$ be a complete Riemannian
manifold with a fixed Riemnnian metric $g$, $\phi\in C^{2}(M)$ and
$d\mu=e^{-\phi}dv$, where $v$ is the Riemannian volume measure on $(M,g)$, and
let

$\displaystyle L=\Delta-\nabla\phi\cdot\nabla$

be the Witten Laplacian on $(M,g)$ with respect to the weighted volume measure
$\mu$. Following [5, 32, 25], we introduce the $m$-dimensional Bakry-Emery
Ricci curvature on $(M,g,\phi)$ by

$\displaystyle
Ric_{m,n}(L):=Ric+\nabla^{2}\phi-{\nabla\phi\otimes\nabla\phi\over m-n},$

where $m\geq n$ is a constant, and $m=n$ if and only if $\phi$ is a constant.
Let $u$ be a positive solution of the heat equation

$\displaystyle\partial_{t}u=Lu,$

and define

$\displaystyle H_{m}(u,t)=-\int_{M}u\log ud\mu-{m\over 2}\left(1+\log(4\pi
t)\right).$

In [28], the author introduced the $W$-entropy for the Witten Laplacian by

$\displaystyle W_{m}(u,t)={d\over dt}(tH_{m}(u,t)),$ (1)

and proved that

$\displaystyle W_{m}(u,t)=\int_{M}\left(t|\nabla
f|^{2}+f-m\right){e^{-f}\over(4\pi t)^{m/2}}d\mu,$ (2)

where $u={e^{-f}\over(4\pi t)^{m/2}}$. Moreover, it was proved in [28] that
the following $W$-entropy formula holds if $u={e^{-f}\over(4\pi t)^{m/2}}$ is
the heat kernel of the Witten Laplacian on complete Riemannian manifolds
satisfying the so-called $\mu$-bounded geometry condition

$\displaystyle{dW_{m}(u,t)\over dt}$ $\displaystyle=$
$\displaystyle-2\int_{M}t\left(\left|\nabla^{2}f-{g\over
2t}\right|^{2}+Ric_{m,n}(L)(\nabla f,\nabla f)\right)ud\mu$ (3)
$\displaystyle\hskip 56.9055pt-{2\over
m-n}\int_{M}t\left({\nabla\phi\cdot\nabla f}+{m-n\over 2t}\right)^{2}ud\mu.$

As a consequence, if $(M,g,\phi)$ satisfies the $\mu$-bounded geometry
condition and

$\displaystyle Ric_{m,n}(L)\geq 0,$

then the $W$-entropy is non-decreasing in time $t$, i.e.,

$\displaystyle{dW_{m}(u,t)\over dt}\leq 0,\ \ \ \ \ \ \ \forall t\geq 0,$

which extends the above result due to Ni [39, 40]. Moreover, under the above
conditions, the author proved that ${d\over dt}W_{m}(u,t)=0$ for some $t>0$ if
and only if $M$ is isometric to $\mathbb{R}^{n}$, $m=n$, $\phi$ is a constant,
$L=\Delta$, and $u$ is the standard Gaussian heat kernel on $\mathbb{R}^{n}$.

When $m\in\mathbb{N}$, consider the warped product metric on $M^{n}\times
S^{m-n}$ defined by

$\displaystyle\widetilde{g}=g_{M}\bigoplus e^{-{2\phi\over m-n}}g_{S^{m-n}}.$

where $S^{m-n}$ is the unit sphere in $\mathbb{R}^{m-n+1}$ with the standard
metric $g_{S^{m-n}}$. By a classical result in Riemaniann geometry, the
quantity $Ric_{m,n}(L)$ is equal to the Ricci curvature of the above warped
product metric $\widetilde{g}$ on $M^{n}\times S^{m-n}$ along the horizontal
vector fields. See [32, 25, 49] and reference therein. Thus, the
$m$-dimensional Bakry-Emery Ricci curvature $Ric_{m,n}(L)$ has a very natural
geometric interpretation. On the other hand, according to [28], we say that
$(M,g,\phi)$ satisfies the $\mu$-bounded geometry condition, if the Riemannian
curvature tensor as well as its first and second order derivatives are
uniformly bounded, $\phi\in C^{4}(M)$ with $\nabla\phi\in C_{b}^{3}(M)$, and
the $\mu$-volume of the geodesic balls of a fixed radius $r_{0}>0$ is
uniformly bounded from below by a positive constant with respect to their
centers, i.e.,

$\displaystyle\mu_{0}:=\inf\limits_{x\in M}\mu(B(x,r_{0}))>0.$ (4)

It is easy to see that the uniformly volume lower bound condition
$(\ref{mulow})$ implies that the total $\mu$-volume of $M$ is infinity if $M$
is non-compact. This excludes the case where $\mu$ is a probability measure on
$M$. Moreover, by Theorem 2.8 in [27], on complete Riemannian manifolds with
$Ric_{m,n}(L)\geq-K$ for some constant $K\geq 0$ and with the uniformly lower
bound condition $(\ref{mulow})$, the following Sobolev inequality holds

$\displaystyle\|f\|^{2}_{2m\over m-2}\leq C\left(\int_{M}|\nabla
f|^{2}d\mu+\int_{M}|f|^{2}d\mu\right),\ \ \forall f\in C_{0}^{\infty}(M).$

Thus, at least from a geometric and analytic point of view, the uniformly
volume lower bound condition $(\ref{mulow})$ is too restrictive for the
validity of the $W$-entropy formula $(\ref{W-1})$ for the Witten Laplacian on
complete Riemannian manifolds. It is very natural to ask the question whether
the $W$-entropy formula $(\ref{W-1})$ remains valid on complete Riemannian
manifolds without assuming the uniformly volume lower bound condition
$(\ref{mulow})$. This is certainly an important issue in view of an extensive
interest on the study of geometry and analysis on complete Riemannian
manifolds with smooth weighted volume measures [5, 6, 16, 25, 28, 29, 38, 49,
48, 36, 37].

The purpose of this paper is to prove the $W$-entropy formula for the Witten
Laplacian on complete Riemannian manifolds with more natural geometric
conditions in which one need not to assume the uniform lower bound condition
$(\ref{mulow})$. Indeed, after the paper [28] was accepted, we observed that
the uniform volume lower bound condition $(\ref{mulow})$ can be removed. See
the Note added in proof of [28]. Our observation is essentially based on the
fact that one can prove some Hamilton type gradient estimates for the
logarithmic of the heat kernel of the Witten Laplacian on complete Riemannian
manifolds on which one need not to assume the uniform volume lower bound
condition $(\ref{mulow})$. In [28], the author obtained these type gradient
estimates on complete Riemannian manifolds satisfying the $\mu$-bounded
geometry condition. In this paper, we prove Hamilton’s Harnack inequality for
all bounded and positive solutions to the heat equation of the Witten
Laplacian and prove the Hamilton type gradient estimates of the logarithmic of
the heat kernel for the Witten Laplacian on complete Riemannian manifolds with
natural geometric conditions. As application, we derive the $W$-entropy
formula and prove the rigidity theorem of the $W$-entropy for the Witten
Laplacian on complete Riemannian manifolds with natural Bakry-Emery curvature
condition.

The rest of this paper is organized as follows. In Section $2$, we state the
main results of this paper. In Section $3$, we prove Hamilton’s Harnack
inequality for the Witten Laplacian on complete Riemanian manifolds. In
Section $4$, we prove the Hamilton type gradient estimates of the logarithmic
heat kernel of the Witten Laplacian using a method of stochastic analysis. In
Section $5$, we prove the $W$-entropy formula and derive the monotonicity and
rigidity theorem of the $W$-entropy for the Witten Laplacian on complete
Riemannian manifolds with natural Bakry-Emery Ricci curvature condition. In
Section $6$, we prove a family of logarithmic Sobolev inequalities on complete
Riemannian manifolds with $\mu$-bounded geometry condition. In Section $7$, we
give an analytic proof of the first order gradient estimate of the logarithmic
heat kernel of the Witten Laplacian operator. In Section $8$ we develop
another probabilistic approach to the Hamilton type Harnack inequality for the
Witten Laplacian on complete Riemanian manifolds.

##  2 Statement of the main results

Let $(M,g)$ be a complete Riemannian manifold, $f\in C^{2}(M)$ and
$d\mu=e^{-f}dv$, where $dv$ denotes the Riemannian volume measure on $(M,g)$.
For all $u,v\in C^{\infty}_{0}(M)$, the following integration by parts formula
holds

$\displaystyle\int_{M}\langle\nabla u,\nabla v\rangle
d\mu=-\int_{M}Luvd\mu=-\int_{M}uLvd\mu,$

where $L=\Delta-\nabla\phi\cdot\nabla$ is the Witten Laplacian on
$(M,g,\phi)$. In [5], Bakry and Emery proved that for all $u\in
C_{0}^{\infty}(M)$,

$\displaystyle L|\nabla u|^{2}-2\langle\nabla u,\nabla
Lu\rangle=2|\nabla^{2}u|^{2}+2(Ric+\nabla^{2}\phi)(\nabla u,\nabla u).$ (5)

The formula $(\ref{BWF})$ can be viewed as a natural extension of the Bochner-
Weitzenböck formula. The quantity $Ric+\nabla^{2}\phi$, which is called in the
literature the Bakry-Emery Ricci curvature on the weighted Riemannian
manifolds $(M,g,\phi)$, and plays as a good substitute of the Ricci curvature
in many problems in comparison geometry and analysis on complete Riemannian
manifolds with smooth weighted volume measures. See [5, 16, 25, 28, 29, 38,
49, 36, 37] and reference therein.

Consider the heat equation

$\displaystyle{\partial u\over\partial t}=Lu.$ (6)

The heat kernel $p_{t}(x,y)$ to the heat equation $(\ref{HEL})$ is defined by

$\displaystyle u(t,x)=\int_{M}u(0,y)p_{t}(x,y)d\mu(y).$

We now state the main results of this paper. The first result of this paper is
the following Hamilton’s Harnack inequality for the heat equation of the
Witten Laplacian on complete Riemannian manifolds.

###### Theorem 2.1

Let $M$ be a complete Riemannian manifold, and $\phi\in C^{2}(M)$. Suppose
that there exists a constant $K\geq 0$ such that

$\displaystyle Ric+\nabla^{2}\phi\geq-K.$

Let $u$ be a positive and bounded solution to the heat equation

$\displaystyle\partial_{t}u=Lu.$

Then for all $x\in M$ and $t>0$,

$\displaystyle|\nabla\log u|^{2}\leq\left({1\over t}+2K\right)\log(A/u),$ (7)

where

$\displaystyle A:=\sup\limits\\{u(t,x):x\in M,t\geq 0\\}.$

As a corollary of Theorem 2.1, we can derive the following gradient estimate
and the Liouville theorem due to Brighton [7]. See also Remark 3.4 below.

###### Corollary 2.2

Let $u$ be a bounded and positive $L$-harmonic function $u$ on $M$. Then,
under the same condition as in Theorem 2.1, we have

$\displaystyle|\nabla\log u|\leq 2K\log(A/(u-\inf\limits u)),$

where $A=\sup\limits u-\inf\limits u$. In particular, if
$Ric+\nabla^{2}\phi\geq 0$, then every bounded $L$-harmonic function must be
constant.

The following two results provide us with the Hamilton type gradient estimates
of the logarithmic heat kernel of the Witten Laplacian on complete Riemannian
manifolds with natural geometric conditions.

###### Theorem 2.3

Let $M$ be a complete Riemannian manifolds, and $\phi\in C^{2}(M)$. Suppose
that there exist some constant $m\geq n$ and $K\geq 0$ such that

$Ric_{m,n}(L)\geq-K.$

Then, for all $T>0$, there exists a constant $C(K,m,n,T)>0$ such that

$\displaystyle|\nabla\log p_{t}(x,y)|\leq C(K,m,n,T)\left({d(x,y)\over
t}+{1\over\sqrt{t}}\right)$

holds for all $x,y\in M$ and $t\in(0,T]$.

###### Theorem 2.4

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold. Suppose that
$\|\nabla^{k}{\rm Riem}\|\leq C_{k}$ for some constant $C_{k}>0$, $0\leq k\leq
N$, $\phi\in C^{N+1}(M)$ with $\nabla\phi\in C_{b}^{N}(M)$, and there exists
some $m\geq n$, $K\geq 0$ such that $Ric_{m,n}(L)\geq-K$. Then, for all $T>0$,
there exists a constant $C_{N}(K,m,n,T)>0$ such that

$\displaystyle|\nabla^{N}\log p_{t}(x,y)|\leq C_{N}(K,m,n,T)\left({d(x,y)\over
t}+{1\over\sqrt{t}}\right)^{N}$

holds for all $x,y\in M$ and $t\in(0,T]$.

Based on Theorem 2.3 and Theorem 2.4, we have the following result on the
$W$-entropy for the Witten Laplacian on complete Riemannian manifold, which
improves a previous result obtained in [28].

###### Theorem 2.5

Let $(M,g)$ be a complete Riemannian manifold. Suppose that $\|\nabla^{k}{\rm
Riem}\|\leq C_{k}$ for some constant $C_{k}>0$, $0\leq k\leq 2$, $\phi\in
C^{4}(M)$ with $\nabla\phi\in C_{b}^{3}(M)$, and there exists some $m\geq n$,
$K\geq 0$ such that $Ric_{m,n}(L)\geq-K$. Let

$u(t,x)={e^{-f}\over(4\pi t)^{m/2}}.$

be the heat kernel of the heat equation

${\partial u\over\partial t}=Lu.$

Then, for all $t>0$, we have the following entropy formula

$\displaystyle{dW_{m}(u,t)\over dt}$ $\displaystyle=$
$\displaystyle-2\int_{M}t\left(\left|\nabla^{2}f-{g\over
2\tau}\right|^{2}+Ric_{m,n}(L)(\nabla f,\nabla f)\right)ud\mu$ $\displaystyle\
\ -{2\over m-n}\int_{M}t\left({\nabla\phi\cdot\nabla f}+{m-n\over
2t}\right)^{2}ud\mu,$

where

$\displaystyle W_{m}(u,t)=\int_{M}\left(t|\nabla
f|^{2}+f-m\right){e^{-f}\over(4\pi t)^{m/2}}d\mu$

is the $W$-entropy of the Witten Laplacian $L=\Delta-\nabla\phi\cdot\nabla$ on
$(M,g,\mu)$.

As a consequence of the above $W$-entropy formula, we have the following
monotonicity and rigidity theorem for the $W$-entropy for the Witten Laplacian
on complete Riemannian manifolds, which also improves the previous result
obtained in [28].

###### Theorem 2.6

Let $(M,g)$ be a complete Riemannian manifold. Suppose that $\|\nabla^{k}{\rm
Riem}\|\leq C_{k}$ for some constant $C_{k}>0$, $0\leq k\leq 2$, $\phi\in
C^{4}(M)$ with $\nabla\phi\in C_{b}^{3}(M)$. Suppose that there exists a
constant $m\geq n$ such that

$\displaystyle Ric_{m,n}(L)\geq 0.$

Then

$\displaystyle{dW_{m}(u,t)\over dt}\leq 0,\ \ \ \forall t\geq 0.$

That is, $W_{m}(u,t)$ is non-increasing in time $t$. Moreover, $W_{m}(u,t)$
attains its minimum at some point $t_{0}>0$, i.e.,

$\displaystyle{dW_{m}(u,t)\over dt}=0$

holds at $t=t_{0}$, if and only if $(M,g)$ is isometric to Euclidean space
$\mathbb{R}^{n}$, $m=n$, $\phi\equiv C$ for a constant $C\in\mathbb{R}$, and

$\displaystyle u(t,x)={e^{-{\|x\|^{2}\over 4t}}\over(4\pi t)^{n/2}},\ \ \ \
\forall x\in\mathbb{R}^{n},t>0.$

The following result says that the uniform $\mu$-volume lower bound condition
$(\ref{mulow})$ is indeed an important condition which implies the validity of
a family of logarithmic Sobolev inequalities on complete Riemannian manifolds
with $\mu$-bounded geometry condition.

###### Theorem 2.7

Let $M$ be a complete Riemannian manifold, and $\phi\in C^{2}(M)$. Suppose
that there exist some constants $m\geq n$ and $K\geq 0$ such that

$\displaystyle Ric_{m,n}(L)\geq-K,$

and the uniform volume lower bound condition $(\ref{mulow})$ holds, i.e.,
there exists a constant $r_{0}>0$ such that

$\displaystyle\mu_{0}=\inf\limits\\{\mu(B(x,r_{0}):x\in M\\}>0.$

Then, for any $\tau>0$, the following logarithmic Sobolev inequality holds:
for all $v\in W^{1,2}(M,\mu)$ with $\int_{M}v^{2}d\mu=1$, we have

$\displaystyle\int_{M}v^{2}\log v^{2}d\mu\leq 4\tau\int_{M}|\nabla
v|^{2}-m\left(1+{1\over 2}\log(4\pi\tau)\right)-\mu(\tau),$

where

$\displaystyle\mu(\tau):=\inf\limits\left\\{W_{m}(u,\tau):\
u={e^{-f}\over(4\pi\tau)^{m/2}},\ \int_{M}ud\mu=1\right\\}$

is a finite number.

###### Remark 2.8

When $m\in\mathbb{N}$, as was pointed out in [28], the quantity

$H_{m}(u,t)=-\int_{M}u\log ud\mu-{m\over 2}\left(1+\log(4\pi t)\right)$

defined in $(\ref{H-E})$ is the difference of the Bolztmann-Shannon-Nash
entropy

$H(u,t)=-\int_{M}u\log ud\mu$

of the time $t$-heat kernel measure $ud\mu$ on $(M,g)$ and the Bolztmann-
Shannon-Nash entropy

$H(\gamma_{m},t)={m\over 2}\left(1+\log(4\pi t)\right)$

of the time $t$-Gaussian measure $\gamma_{m}(\cdot,t)$ on $\mathbb{R}^{m}$,
where

$d\gamma_{m}(x,t)={e^{-{\|x\|^{2}\over 4t}}\over(4\pi t)^{m/2}}dx.$

This observation together with the definition formula $(\ref{W-H})$, i.e.,

$\displaystyle W_{m}(u,t)={d\over dt}(tH_{m}(u,t)),$

gives a probabilistic interpretation of the $W$-entropy functional
$(\ref{W-E})$ for the Witten Laplacian on complete Riemannian manifolds. See
also [28] for the probabilistic interpretation of Perelman’s $W$-entropy for
the Ricci flow.

###### Remark 2.9

Theorem 2.3 and Theorem 2.4 are improved versions of Theorem 4.1 and Theorem
4.2 due to the author [28], where one need to assume the additional condition
$(\ref{mulow})$ holds. When M is a compact Riemannian manifold and $\phi$ is a
constant, Theorem 2.3 and Theorem 2.4 are due to Hamilton [18], Sheu [42], Hsu
[20] and Stroock-Turesky [44]. In the case where $L=\Delta$ is the Laplace-
Beltrami operator on complete Riemannian manifolds with bounded geometry
condition (in particular, the uniform volume lower bound condition
$(\ref{mulow})$ holds for $\mu=\nu$) and for small time $t>0$, Theorem 2.3 is
due to Engoulatov [15]. In [22, 13], the authors proved the gradient estimates
in Theorem 2.3 for $L=\Delta$ on complete Riemannin manifolds with non-
negative Ricci curvature.

###### Remark 2.10

In a very recent paper [30], Songzi Li and the author gave a new proof of the
$W$-entropy formula on complete Riemannian manifolds by using the warped
metric product

$\displaystyle\widetilde{g}=g_{M}\bigoplus e^{-{2\phi\over m-n}}g_{N}.$

on $M\times N$, where $(N,g_{N})$ is any fixed compact Riemannian manifold of
dimension $m-n$. This gives a natural geometric interpretation of the last
term appeared in $(\ref{W-1})$. See Remark $2.2$ in [30]. Moreover, we
extended the $W$-entropy formula to the heat equation of the time dependent
Witten Laplacian on compact Riemannian manifolds on which the metrics
$\\{g(t),t\in[0,T]\\}$ and the potential functions $\\{\phi(t),t\in[0,T]\\}$
evolve along the $m$-dimensional Perelman Ricci flow and the conjugate heat
equation

$\displaystyle{\partial g\over\partial t}$ $\displaystyle=$
$\displaystyle-2\left(Ric+\nabla^{2}\phi-{\nabla\phi\otimes\nabla\phi\over
m-n}\right),$ $\displaystyle{\partial\phi\over\partial t}$ $\displaystyle=$
$\displaystyle-R-\Delta\phi+{|\nabla\phi|^{2}\over m-n}.$

In this case, without assuming $Ric_{m,n}(L)\geq 0$, we prove in [30] that the
$W$-entropy for the positive solution of the forward heat equation

$\partial_{t}u=Lu$

of the time dependent Witten Laplacian $L=\Delta-\nabla\phi\cdot\nabla$, is
non-increasing in time $t\in[0,T]$.

##  3 Hamilton’s Harnack inequality

In this section we prove Hamilton’s Harnack inequality for the Witten
Laplacian.

###### Theorem 3.1

(i.e., Theorem 2.1) Let $u$ be a bounded and positive solution of the heat
equation $\partial_{t}u=Lu$. Suppose that $Ric+\nabla^{2}\phi\geq-K$. Then for
all $T>0$, we have

$\displaystyle|\nabla\log u(T,x)|^{2}\leq\left({1\over
T}+2K\right)\log\left[{A\over u(T,x)}\right],$

where

$\displaystyle A:=\sup\limits\\{u(t,x):x\in M,t\in[0,T]\\}.$

Proof. In compact case, we can prove it by the same argument as used by
Hamilton [18] for the Laplace-Beltrami. Indeed, a direct calculation yields

$\displaystyle(\partial_{t}-L){|\nabla u|^{2}\over u}$ $\displaystyle=$
$\displaystyle-{2\over u}|\nabla^{2}u-u^{-1}\nabla u\otimes\nabla
u|^{2}-2u^{-1}(Ric+\nabla^{2}\phi)(\nabla u,\nabla u)$ $\displaystyle\leq$
$\displaystyle 2Ku^{-1}|\nabla u|^{2}.$

Moreover

$\displaystyle(\partial_{t}-L)(u\log(A/u))={|\nabla u|^{2}\over u}.$

Let $\phi(t)={t\over 2Kt+1}$. Then

$\displaystyle\phi^{\prime}(t)+2K\phi(t)\leq 1.$

Define

$\displaystyle h(x,t):=\phi(t){|\nabla u|^{2}\over u}-u\log(A/u).$

We have

$\displaystyle(\partial_{t}-L)h$ $\displaystyle=$
$\displaystyle\phi^{\prime}(t){|\nabla u|^{2}\over
u}+\phi(t)(\partial_{t}-L){|\nabla u|^{2}\over u}-(\partial_{t}-L)u\log(A/u)$
$\displaystyle\leq$ $\displaystyle\phi^{\prime}(t){|\nabla u|^{2}\over
u}+2K\phi(t){|\nabla u|^{2}\over u}-{|\nabla u|^{2}\over u}$ $\displaystyle=$
$\displaystyle(\phi^{\prime}(t)+2K\phi(t)-1){|\nabla u|^{2}\over u}$
$\displaystyle\leq$ $\displaystyle 0.$

Note that

$h(x,0)=-u\log(A/u)(x,0)\leq 0.$

By the maximum principle, we have $h(x,t)\leq 0$ for all time $t>0$ and $x\in
M$. This proves Theorem 3.1 in compact case. In general case of complete non-
compact Riemannian manifolds with $Ric(L)\geq-K$, we can give a probabilistic
proof to Hamilton’s Harnack inequality as follows. Indeed, on any complete
Riemannian manifold $M$ with $Ric+\nabla^{2}\phi\geq K$, a previous result due
to Bakry [3] says that the $L$-diffusion process on $M$ has infinity lifetime.
Let $X_{t}$ be the $L$-diffusion process on $M$ starting from $X_{0}=x$.
Applying Itô’s formula to $h(X_{t},T-t)$, $t\in[0,T]$, we have

$\displaystyle h(X_{t},T-t)=h(X_{0},T)+\int_{0}^{t}\nabla h(X_{s},T-s)\cdot
dW_{s}+\int_{0}^{t}\left(L-{\partial\over\partial t}\right)h(X_{s},T-s)ds,$

where the second term in the right hand side is the Itô’s stochastic integral
with respect to a $M$-valued Brownian motion $\\{W_{s},s\in[0,t]\\}$. In
particular, taking $t=T$, we obtain

$\displaystyle h(X_{T},0)=h(X_{0},T)+\int_{0}^{T}\nabla h(X_{s},T-s)\cdot
dW_{s}+\int_{0}^{T}\left(L-{\partial\over\partial t}\right)h(X_{s},T-s)ds.$

Taking the expectation on both sides, the martingale property of Itô’s
integral implies that

$\displaystyle
E\left[h(X_{T},0)\right]=h(x,T)+E\left[\int_{0}^{T}\left(L-{\partial\over\partial
t}\right)h(X_{s},T-s)ds\right]\geq h(x,T).$

As $h(y,0)\leq 0$ for all $y\in M$, we derive that $h(x,T)\leq 0$ for all
$T>0$ and $x\in M$. This finishes the proof of Theorem 3.1 on complete non-
compact Riemannian manifolds. $\square$

An alternative proof of Theorem 3.1. By [4, 6, 48], we know that
$Ric(L)=Ric+\nabla^{2}\phi\geq-K$ holds if and only the following version of
logarithmic Sobolev inequality holds: for all $T>0$, $f\in C_{b}(M)$ with
$f>0$,

$\displaystyle{|\nabla P_{T}f|^{2}\over P_{T}f}\leq{2K\over
1-e^{-2KT}}\left(P_{T}(f\log f)-P_{T}f\log P_{T}f\right).$

Replacing $f$ by $P_{t}f$ and using the fact that $0<P_{t}f(x)\leq A$ for all
$x\in M$, one has

$\displaystyle{|\nabla P_{T+t}f|^{2}\over P_{T+t}f}\leq{2K\over
1-e^{-2KT}}\left(P_{T}(P_{t}f\log A)-P_{T+t}f\log P_{T+t}f\right).$

Thus

$\displaystyle|\nabla\log P_{T+t}f|^{2}\leq{2K\over
1-e^{-2KT}}\log(A/P_{T+t}f),\ \ \forall\ t>0.$

Taking $t\rightarrow 0^{+}$ we have

$\displaystyle|\nabla\log P_{T}f|^{2}$ $\displaystyle\leq$
$\displaystyle{2K\over 1-e^{-2KT}}\log(A/P_{T}f)$ $\displaystyle\leq$
$\displaystyle 2K\left(1+{1\over 2KT}\right)\log(A/P_{T}f)$
$\displaystyle\leq$ $\displaystyle\left(2K+{1\over T}\right)\log(A/P_{T}f).$

The proof of Theorem 3.1 is completed. $\square$

###### Remark 3.2

As was pointed out in Section 1, Theorem 2.1 was first proved by Hamilton [18]
for the heat equation of the Laplace-Beltrami operator on compact Riemannian
manifolds and was extended to complete Riemannian manifolds with Ricci
curvature bounded from below by Kotschwar [22]. For a probabilistic proof of
Hamilton’s Harnack inequality for $L=\Delta$ on compact or complete Riemannian
manifolds, see [1]. The local version of Theorem 2.1 for the Laplace-Beltrami
operator was proved by Souplet-Zhang [43]. In [2] (Theorem 5.1), Arnaudon,
Tahlmaier and Wang extended the local estimate of Souplet-Zhang [43] to the
heat equation of the Laplacian with general drift $L=\Delta+Z$, where $Z$ is
not necessary of gradient form $Z=\nabla\phi$ and the Laplacian comparison
theorem for $L=\Delta+Z$ is essentially used. Taking $R\rightarrow\infty$ in
[2], we can derive the following estimate for the bounded and positive
solution of the heat equation $\partial_{t}u=Lu$: there exists a constant
$C>0$, which depends only on $n={\rm dim}M$, such that for all $x\in M$ and
$t>0$, it holds

$\displaystyle|\nabla\log u|^{2}\leq C\left({1\over
t}+K\right)\left[1+\log(A/u)\right].$ (8)

where $A:=\sup\limits\\{u(t,x):x\in M,t\geq 0\\}$. Note that the Harnack
inequality $(\ref{Ham})$ is stronger than $(\ref{ATMW})$. See also Section 8
for another probabilistic proof of the Hamilton type Harnack inequality for
the Witten Laplacian on complete Riemannian manifolds.

As a corollary of the above theorem, we can derive the following gradient
estimate and the Liouville theorem due to Brighton [7] for bounded
$L$-harmonic functions.

###### Corollary 3.3

(i.e., Corollary 2.2)) Suppose that $Ric(L)=Ric+\nabla^{2}\phi\geq-K$ for some
constant $K\geq 0$. Then every bounded $L$-harmonic function $u$ satisfies

$\displaystyle|\nabla\log u|\leq 2K\log(A/(u-\inf\limits u)),$ (9)

where $A=\sup\limits u-\inf\limits u$. In particular, if
$Ric(L)=Ric+\nabla^{2}\phi\geq 0$, then every bounded $L$-harmonic function
must be constant.

Proof. Applying $(\ref{E31b})$ to $u(t,x)=u(x)-\inf\limits u$, $x\in M$,
$t>0$, we have

$\displaystyle|\nabla\log u(x)|^{2}\leq\left(2K+{1\over
T}\right)\log\left({A/(u(x)-\inf\limits u)}\right),\ \ \forall T>0.$

Taking $T\rightarrow\infty$, we derive $(\ref{grad-1})$. In particular, if
$K=0$, we have $|\nabla\log u|=0$. This finishes the proof of Corollary 3.3.
$\square$

###### Remark 3.4

In the case $L=\Delta$, Yau [46] proved that, on complete Riemannian manifolds
with Ricci curvature bounded below by $-K$, i.e., $Ric\geq K$, every harmonic
function which is bounded below by a constant satisfies the gradient estimate

$\displaystyle|\nabla\log u|\leq\sqrt{(n-1)K}(u-\inf\limits u).$

In particular, Yau [46] proved the Strong Liouville theorem holds: On complete
Riemannian manifolds with non-negative Ricci curvature, every positive (and
hence bounded) harmonic function must be constant. See also [45]. In [25], the
author extended Yau’s gradient estimate and Strong Liouville theorem to the
$L$-harmonic functions on complete Riemnnian manifolds via the Bakry-Emery
Ricci curvature: Suppose that $Ric_{m,n}(L)\geq-K$ for some constant $K\geq
0$. Then every $L$-harmonic function $u$, which is bounded from below,
satisfies the following gradient estimate

$\displaystyle|\nabla\log u|\leq\sqrt{(m-1)K}(u-\inf\limits u).$

In particular, if $Ric_{m,n}(L)\geq 0$, then all positive (and hence bounded)
$L$-harmonic functions must be constant. In [26], the local gradient estimates
was proved for $L$-harmonic functions: If $Ric_{m,n}(L)\geq-K$, then every
positive $L$-harmonic function $u$ on $M$ satisfies: for all $o\in M$, and
$R>0$, we have

$\displaystyle|\nabla\log
u(x)|\leq\sqrt{(m-1)K}u(x)+C(K,m)(R^{-1/2}+R^{-3/2}),\ \ \forall x\in B(o,R).$

In [7], Brighton proved that, on complete Riemannian manifolds with
$Ric+\nabla^{2}\phi\geq 0$, every bounded $L$-harmonic function must be
constant, i.e., Corollary 2.2. Our method for the proof of Corollary 2.2 is
different from [7]. For more recent results, see [36, 37].

##  4 Gradient estimates of the heat kernel

In this section we use a probabilistic approach to prove Theorem 2.3 and
Theorem 2.4. Our proof is inspired by previous works initialed by Sheu [42],
and developed by Hsu [20] and Engoulatov [15]. An alternative proof of Theorem
2.3 is given in Section $8$ below.

Let $O(M)$ be the orthogonal frame bundle over $M$, $\pi:O(M)\rightarrow M$
the canonical projection map. Let $H_{1},\ldots,H_{n}$ be the canonical
horizonal vector fields on $O(M)$. Let $B_{t}$ be the standard Brownian motion
on $\mathbb{R}^{n}$. Following Malliavin [33, 34], we define the horizontal
$L$-diffusion process $U_{t}$ on $O(M)$ by the following Stratonovich SDE on
$O(M)$:

$\displaystyle dU_{t}=\sum\limits_{i=1}^{n}H_{i}(U_{t})\circ
dB^{i}_{t}-\nabla^{H}\phi(U_{t})dt,$ (10)

where $\nabla^{H}\phi$ denotes the horizontal lift of the gradient vector
field $\nabla\phi$ on $O(M)$, which is the unique horizontal vector field on
$O(M)$ such that

$\displaystyle\pi_{*}(\nabla^{H}\phi)=\nabla\phi.$

Let

$\displaystyle X_{t}=\pi(U_{t}),\ \ \forall t>0.$

Then $X_{t}$ is a diffusion process on $M$ with infinitesimal generator $L$.
Moreover, we have

$\displaystyle dX_{t}=U_{t}\circ dB_{t}-\nabla\phi(X_{t})dt.$ (11)

Let $\Delta_{O(M)}=\sum\limits_{i=1}^{n}H_{i}^{2}$ be the horizontal Laplace-
Beltrami on $O(M)$. The horizontal Witten Laplacian is deined by

$L_{O(M)}=\Delta_{O(M)}-\nabla^{H}\phi\cdot\nabla^{H}.$

Fix $T>0$. Let $\mathbb{P}_{x,y,T}$ be the conditional $L$-diffusion processes
starting from $x$ and ending at $y$ t time $T$, i.e.,

$\displaystyle\mathbb{P}_{x,y,T}(\cdot)=\mathbb{P}_{x}(\cdot|X_{T}=y).$

Let

$\displaystyle J(t,u)=\log p_{T-t}(\pi u,y),\ \ \ \forall u\in
O(M),t\in[0,T].$

By the Grirsanov transformation, we have

$\displaystyle\left.{d\mathbb{P}_{x,y,T}\over
d\mathbb{P}_{x}}\right|_{\mathcal{F}_{t}}={p_{T-t}(X_{t},y)\over
p_{T}(x,y)}=\exp\\{J(t,U_{t})-J(0,U_{0})\\},\ \ \ \forall t<T.$

where $\mathcal{F}_{t}=\sigma(X_{s},s\leq t)$ is the natural $\sigma$-filed
generated by $X_{t}$. By Itô’s formula, we have

$\displaystyle dJ(t,U_{t})={\partial J\over\partial
t}(t,U_{t})dt+\sqrt{2}\langle\nabla^{H}J(t,U_{t}),dB_{t}\rangle+L_{O(M)}J(t,U_{t})dt,$

Note that $J(t,u)$ satisfies the Hamilton-Jacobi equation

$\displaystyle{\partial J\over\partial t}+L_{O(M)}J+|\nabla^{H}J|^{2}=0,$ (12)

which yields

$\displaystyle\left.{d\mathbb{P}_{x,y,T}\over
d\mathbb{P}_{x}}\right|_{\mathcal{F}_{t}}=\exp\left[\int_{0}^{t}\langle\sqrt{2}\nabla^{H}J(s,U_{s}),dB_{s}\rangle-\int_{0}^{T}|\nabla^{H}J(s,U_{s})|^{2}ds\right],\
\ \ \forall t<T.$

It follows, by Girsanov’s theorem, that the process

$\displaystyle b_{t}=B_{t}-\sqrt{2}\int_{0}^{t}\nabla^{H}J(s,U_{s})ds,\ \ \
s\in[0,T),$ (13)

is a Brownian motion under $\mathbb{P}_{x,y,T}$. Substituting $(\ref{b})$ into
$\ref{U})$ and $(\ref{X})$, we have the following lemma.

###### Lemma 4.1

The conditional $L$-diffusion process $X_{t}$ on $M$ staring from $x$ with
terminal end $y$ at the time $T$, and its horizontal lift $U_{t}$, are the
solutions of the following Stratonovich SDEs:

$\displaystyle dU_{t}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}H_{i}(U_{t})\circ\left[\sqrt{2}db_{t}^{i}+2H_{i}\log
p_{T-t}(U_{t},y)dt\right]-\nabla^{H}\phi(U_{t})dt,$ $\displaystyle dX_{t}$
$\displaystyle=$ $\displaystyle U_{t}\circ\left[\sqrt{2}db_{t}+2\nabla^{H}\log
p_{T-t}(U_{t},y)dt\right]-\nabla\phi(X_{t})dt,$

where $b_{t}$ is a standard Brownian motion on $\mathbb{R}^{n}$ under
$\mathbb{P}_{x,y,T}$.

###### Proposition 4.2

For all $0<t<T$, we have

$\displaystyle\mathbb{E}[d|\nabla^{H}J(t,U_{t})|^{2}]=2\mathbb{E}\left[|\nabla^{H}\nabla^{H}J|^{2}+(Ric+\nabla^{2}\phi)(\nabla^{H}J,\nabla^{H}J)\right].$
(14)

Proof. By Lemma 4.1 and Itô’s formula, we have

$\displaystyle
d|\nabla^{H}J(t,U_{t})|^{2}=\langle\nabla^{H}|\nabla^{H}J|^{2},\sqrt{2}db_{t}+2\nabla^{H}Jdt\rangle+(\partial_{t}+L_{O(M)})|\nabla^{H}J|^{2}dt.$

By the generalized Bochner-Weitzenböck formula, the action of the horizontal
Witten Laplacian $L_{O(M)}$ on $|\nabla^{H}J|^{2}$ is given by

$\displaystyle
L_{O(M)}|\nabla^{H}J|^{2}=2\langle\nabla^{H}L_{O(M)}J,\nabla^{H}J\rangle+2|\nabla^{H}\nabla^{H}J|^{2}+2(Ric+\nabla^{2}\phi)(\nabla^{H}J,\nabla^{H}J).$

On the other hand, using the Hamilton-Jacobi equation $(\ref{HJJ})$, we have

$\displaystyle\partial_{t}|\nabla^{H}J|^{2}=2\langle\partial_{t}\nabla^{H}J,\nabla^{H}J\rangle=-2\langle
L_{O(M)}\nabla^{H}J,\nabla^{H}J\rangle-2\langle\nabla^{H}|\nabla^{H}J|^{2},\nabla^{H}J\rangle.$

Combining this with the previous calculation, we have

$\displaystyle
d|\nabla^{H}J(t,U_{t})|^{2}=\langle\nabla^{H}|\nabla^{H}J|^{2},\sqrt{2}db_{t}\rangle+2\left[|\nabla^{H}\nabla^{H}J|^{2}+(Ric+\nabla^{2}\phi)(\nabla^{H}J,\nabla^{H}J)\right]dt.$

Taking expectation with respect to $\mathbb{E}_{x,y,T}$, we complete the proof
of Proposition 4.2. $\square$

###### Proposition 4.3

Suppose that $Ric+\nabla^{2}\phi\geq-K$. Then for all $T>0$, we have

$\displaystyle|\nabla\log p_{T}(x,y)|^{2}\leq 2\left({1\over
T}+K\right)\mathbb{E}\left[\log{p_{T/2}(X_{T/2},y)\over p_{T}(x,y)}\right].$
(15)

Proof. Based on Proposition 4.2, the proof of Proposition 4.3 is similar to
the argument used in [20, 15]. For the convenience of the reader, we give the
detail here. Under the condition $Ric+\nabla^{2}\phi\geq-K$, integrating
$(\ref{E1})$ from $0$ to $t$, we have

$\displaystyle\mathbb{E}[|\nabla^{H}J(t,U_{t})|^{2}]-|\nabla^{H}J(0,U_{0})|^{2}\geq-2KE\left[\int_{0}^{t}|\nabla^{H}J(s,U_{s})|^{2}ds\right].$

Integrating again with respect to $t$ we can obtain

$\displaystyle{T\over
2}|\nabla^{H}J(0,U(0)|^{2}\leq(1+KT)\mathbb{E}\left[\int_{0}^{T/2}|\nabla^{H}J(s,U_{s})|^{2}ds\right].$
(16)

By Itô’s formula and the Hamilton-Jacobi equation $(\ref{HJJ})$, we have

$\displaystyle dJ(t,U_{t})$ $\displaystyle=$
$\displaystyle(\partial_{t}+L_{O(M)})Jdt+\langle\sqrt{2}\nabla^{H}J,db_{t}+\sqrt{2}\nabla^{H}Jdt\rangle$
$\displaystyle=$
$\displaystyle\langle\sqrt{2}\nabla^{H}J,db_{t}\rangle+|\nabla^{H}J|^{2}dt.$

Taking expectation with respect to $\mathbb{E}_{x,y,T}$ and integrating from
$t=0$ to $t=T/2$, we get

$\displaystyle\mathbb{E}[J(T/2,U_{T/2})]-J(0,U_{0})=\mathbb{E}\left[\int_{0}^{T/2}|\nabla^{H}J|^{2}dt\right].$
(17)

Combining $(\ref{E2})$ with $(\ref{E4})$, we finish the proof of $(\ref{E3})$.
$\square$

###### Proposition 4.4

Suppose that there exist some constants $m\geq n$, $m\in\mathbb{N}$ and $K\geq
0$ such that $Ric_{m,n}(L)\geq-K$. Then, for any small $\varepsilon>0$, there
exist some constants $C_{i}=C_{i}(m,n,K,\varepsilon)>0$, $i=1,2$, such that
for all $x,y\in M$ and $t>0$,

$\displaystyle p_{t}(x,y)$ $\displaystyle\leq$
$\displaystyle{C_{1}\over\mu(B_{y}(\sqrt{t}))}\exp\left(-{d^{2}(x,y)\over
4(1+\varepsilon)t}+\alpha\varepsilon Kt\right)$ (18) $\displaystyle\hskip
28.45274pt\times\left({d(x,y)+\sqrt{t}\over\sqrt{t}}\right)^{m/2}\exp\left({\sqrt{(m-1)K}d(x,y)\over
2}\right),$

where $\alpha$ is a constant depending only on $m$, and

$\displaystyle p_{t}(x,y)\geq
C_{2}e^{-(1+\varepsilon)\lambda_{K,m}t}\mu^{-1}(B_{y}(\sqrt{t}))\exp\left(-{d^{2}(x,y)\over
4(1-\varepsilon)t}\right)\left[{\sqrt{K}d(x,y)\over\sinh\sqrt{K}d(x,y)}\right]^{m-1\over
2},$ (19)

where

$\lambda_{K,m}={(m-1)^{2}K\over 8}.$

Proof. The lower bound estimate $(\ref{heat-lower})$ has proved in [28]. By
Theorem 5.4 in [25], for all $\varepsilon>0$, there exists a constant
$C_{3}=C_{3}(m,n,K,\varepsilon)>0$

$\displaystyle
p_{t}(x,y)\leq{C_{3}\over\sqrt{\mu(B_{x}(\sqrt{t}))\mu(B_{y}(\sqrt{t}))}}\exp\left(-{d^{2}(x,y)\over
4(1+\varepsilon)t}+\alpha\varepsilon Kt\right).$ (20)

Similarly to [25] (Step 2 of p. 1324), using the Bishop-Gromov relative volume
comparison theorem [38, 25], we have

$\displaystyle\mu(B_{y}(\sqrt{t}))$ $\displaystyle\leq$
$\displaystyle\mu\left(B_{x}(d(x,y)+\sqrt{t})\setminus
B_{x}(d(x,y)-\sqrt{t})\right)$ $\displaystyle\leq$
$\displaystyle\mu(B_{x}(\sqrt{t})){V_{m,K}(d(x,y)+\sqrt{t})-V_{m,K}(d(x,y)-\sqrt{t})\over
V_{m,K}(\sqrt{t})}$ $\displaystyle\leq$
$\displaystyle\mu(B_{x}(\sqrt{t})){V_{m,K}(d(x,y)+\sqrt{t})\over
V_{m,K}(\sqrt{t})},$

where $V_{m,K}(r)$ denotes the volume of geodesic balls of radius $r$ in the
$m$-dimensional hyperbolic space form $\mathbb{H}^{m}(K)$ of constant
sectional curvature $K/m-1$. Using again the Bishop-Gromov volume comparison
theorem on $\mathbb{H}^{m}(K)$, we have

$\displaystyle{\mu(B_{y}(\sqrt{t}))\over\mu(B_{x}(\sqrt{t}))}$
$\displaystyle\leq$ $\displaystyle{V_{m,K}(d(x,y)+\sqrt{t})\over
V_{m,K}(\sqrt{t})}$ (21) $\displaystyle\leq$
$\displaystyle\left({d(x,y)+\sqrt{t}\over\sqrt{t}}\right)^{m}\exp[\sqrt{(m-1)K}d(x,y)].$

By $(\ref{heat-upp-2})$ and $(\ref{vol-mu1})$, we can obtain the upper bound
estimate $(\ref{heat-upp})$. $\square$

Proof of Theorem 2.3. By Proposition 4.4, we have

$\displaystyle{p_{t/2}(X_{t/2},y)\over p_{t}(x,y)}$ $\displaystyle\leq$
$\displaystyle
C_{1}(K,m,T){\mu(B_{y}(\sqrt{t}))\over\mu(B_{y}(\sqrt{t/2}))}\exp\left({d^{2}(x,y)\over
4(1-\varepsilon)t}+{\sqrt{(m-1)K}d(X_{t/2},y)\over 2}\right)$ (22)
$\displaystyle\hskip
56.9055pt\times\left({d(X_{t/2},y)+\sqrt{t}\over\sqrt{t}}\right)^{m/2}\left[{\sqrt{K}d(x,y)\over\sinh\sqrt{K}d(x,y)}\right]^{-{m-1\over
2}}.$

The Bishop-Gromov volume comparison theorem ([38, 25]) implies

$\displaystyle{\mu(B_{y}(\sqrt{t}))\over\mu(B_{y}(\sqrt{t/2}))}\leq
2^{m/2}\exp(\sqrt{m-1}Kt)\leq 2^{m/2}\exp(\sqrt{m-1}KT).$ (23)

Using $(\ref{E5})$, $(\ref{E6})$ and the elementary inequality $x/\sinh x\geq
e^{-x}$, we obtain

$\displaystyle\log\left[{p_{t/2}(X_{t/2},y)\over p_{t}(x,y)}\right]$
$\displaystyle\leq$ $\displaystyle
C_{2}(K,m,T)\left[1+{d(X_{t/2},y)\over\sqrt{t}}+{d^{2}(x,y)\over
t}+d(X_{t/2},y)+d(x,y)\right]$ $\displaystyle\leq$ $\displaystyle
C_{3}(K,m,T)\left[1+{d^{2}(X_{t/2},y)\over t}+{d^{2}(x,y)\over
t}+d(X_{t/2},y)+d(x,y)\right].$

By Itô’s formula, we can prove that (cf. [28])

$\displaystyle\mathbb{E}_{x,y,T}[d^{k}(x_{t/2},y)]\leq(d^{k}(x,y)+1)e^{C_{4}(K,m)t},\
\ \ \forall t>0,\ k=1,2.$

Taking expectation with respect to $\mathbb{E}_{x,y,T}$ in $(\ref{AE1})$, we
have

$\displaystyle\mathbb{E}_{x,y,T}\left[\log{p_{t/2}(X_{t/2},y)\over
p_{t}(x,y)}\right]\leq C_{5}(K,m,T)\left[1+{d^{2}(x,y)+1\over
t}+d(x,y)\right].$

By Proposition 4.3, we have

$\displaystyle|\nabla\log p_{t}(x,y)|^{2}$ $\displaystyle\leq$ $\displaystyle
C_{6}(K,m,T)\left[{1\over t}+{d^{2}(x,y)+1\over t^{2}}+{d(x,y)\over t}\right]$
$\displaystyle\leq$ $\displaystyle C_{7}(K,m,T)\left[{d(x,y)\over
t}+{1\over\sqrt{t}}\right]^{2}.$

The proof of Theorem 2.3 is completed. $\square$

Proof of Theorem 2.4. The proof is similarly to Sheu [42] and Hsu [20]. Let
$I=\\{i_{1},\ldots,i_{N}\\}$ ,and denote $H_{I}=H_{i_{1}}\ldots H_{i_{N}}$,
then by Lemma 4.1 and repeatedly using Itô’s formula, we have

$\displaystyle
dH_{I}J=\langle\nabla^{H}H_{I}J,\sqrt{2}db_{t}\rangle+\left[\partial_{t}H_{I}J+L_{O(M)}H_{I}J+2\langle\nabla^{H}H_{I}J,\nabla^{H}J\rangle\right]dt.$
(24)

Substituting the Hamilton-Jacobi equation $(\ref{HJJ})$ into $(\ref{E7})$, we
have

$\displaystyle
dH_{I}J=\langle\nabla^{H}H_{I}J,\sqrt{2}db_{t}\rangle+(F_{I}+G_{I})dt.$ (25)

where

$\displaystyle
F_{I}=[L_{O(M)},H_{I}]J=[\Delta_{O(M)},H_{I}]J-[\nabla^{H}\phi,H_{I}]J,$

and

$\displaystyle G_{I}=2\langle\nabla^{H}H_{I}J,\nabla^{H}J\rangle-
H_{I}|\nabla^{H}J|^{2}.$

By $(\ref{E4})$ and $(\ref{E5})$ , we have

$\displaystyle\mathbb{E}\left[\int_{0}^{T/2}|\nabla^{H}J(t,U_{t})|^{2}dt\right]\leq
CT\left[{d(x,y)\over T}+{1\over\sqrt{T}}\right]^{2},$ (26)

and by Theorem 2.3, we have

$\displaystyle|H_{i}J(0,U_{0})|\leq C\left[{d(x,y)\over
T}+{1\over\sqrt{T}}\right].$ (27)

Note that

$\displaystyle[\nabla^{H}\phi,H_{I}]J=\sum\limits_{k=1}^{n}[(H_{k}\phi)H_{k},H_{I}]J=\sum\limits_{k=1}^{n}(H_{k}\phi[H_{k},H_{I}]J-H_{I}H_{k}\phi
H_{k}J),$

and by Cartan’s structure theorem, we have

$\displaystyle[H_{i},H_{j}]=V_{ij},\ [H_{i},V_{jk}]=\Omega_{jk}^{il}H_{l},\
[V_{ij},V_{kl}]=c_{ij,kl}^{ab}V_{ab},$

where $V_{ij}$ are the vertical fields on $O(M)$, $\Omega$ is the
$o(n)$-valued curvature form on $O(M)$, and $c_{ij,kl}^{ab}$ are the structure
constants of $o(n)$. The assumption of Theorem 2.4 says that the
$L^{\infty}$-norms $\|\nabla^{k}\Omega\|_{\infty}$ are bounded for all
$k=0,1,\ldots,N$, and the $L^{\infty}$-norms $\|\nabla^{k}\phi\|_{\infty}$ are
bounded for $k=1,\ldots,N+1$. This yields

$\displaystyle|F_{I}|$ $\displaystyle\leq$ $\displaystyle
C_{1}|\nabla^{H}J|+\ldots+C_{1}|(\nabla^{H})^{|I|}J|,$ (28)
$\displaystyle|G_{I}|$ $\displaystyle\leq$ $\displaystyle
C_{1}|\nabla^{H}J|+\ldots+C_{1}|(\nabla^{H})^{|I|}J|+C_{1}|(\nabla^{H})^{|I|}J|^{2},$
(29)

where $C_{1}>0$ is a constant depending only on
$\|\nabla^{i}\Omega\|_{\infty}$ and $\|\nabla^{i}\phi\|_{\infty}$,
$i=1,\ldots,|I|$. By induction, and using $(\ref{E8})$, $(\ref{E9})$,
$(\ref{E10})$, $(\ref{F})$ and $(\ref{G})$, we can prove that for all $I$ with
$2\leq|I|\leq N$,

$\displaystyle\mathbb{E}\left[\int_{0}^{T/2}|H_{I}J(t,U_{t})|^{2}dt\right]\leq
C_{2}\left[{d(x,y)\over T}+{1\over\sqrt{T}}\right]^{2(|I|-1)},$ (30)

and for all $I$ with $|I|\leq N$,

$\displaystyle|H_{I}J(0,U_{0})|\leq C_{3}\left[{d(x,y)\over
T}+{1\over\sqrt{T}}\right]^{|I|}.$ (31)

where $C_{2}$ and $C_{3}$ are two constants depending on $K,m,n,T$ and
$C_{1}$. The proof of Theorem 2.4 is completed. For the detail of the above
induction argument for $L=\Delta$ on compact Riemannian manifolds, see Hsu
[20]. $\square$

##  5 Proofs of Theorem 2.5 and Theorem 2.6

Let

$\displaystyle H(u)=-\int_{M}u\log ud\mu.$

be the Boltzmann-Shannon-Nash entropy of the heat equation of the Witten
Laplacian on $(M,g,\mu)$.

To prove Theorem 2.5 and Theorem 2.6, we need the following entropy
dissipation formulas for the Witten Laplacian on complete Riemannian
manifolds.

###### Theorem 5.1

([28]) Let $M$ be a complete Riemannian manifold, and $\phi\in C^{2}(M)$.
Suppose that there exists a constant $K\geq 0$ such that

$\displaystyle Ric+\nabla^{2}\phi\geq-K.$

Then, for any positive solution to the heat equation

$\displaystyle\partial_{t}u=Lu$

with $u(\cdot,0)\in L^{1}(\mu)\cap L^{4}(d(\cdot,o)d\mu)$, where $o\in M$ is
any fixed point, we have

$\displaystyle{d\over dt}H(u)=-\int_{M}Lu\log ud\mu.$ (32)

###### Theorem 5.2

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold. Suppose that
$\|\nabla^{k}{\rm Riem}\|\leq C_{k}$ for some constant $C_{k}>0$, $0\leq k\leq
3$, $\phi\in C^{4}(M)$ with $\nabla\phi\in C_{b}^{3}(M)$, and there exists
some $m\geq n$, $K\geq 0$ such that $Ric_{m,n}(L)\geq-K$. Then, for the
fundamental solution of the heat equation $\partial_{t}u=Lu$, we have

$\displaystyle{d^{2}\over dt^{2}}H(u)=-\int_{M}\left({|Lu|^{2}\over
u}-\langle\nabla Lu,{\nabla u\over u}\rangle\right)d\mu,$ (33)

and

$\displaystyle{d^{2}\over dt^{2}}H(u)=-2\int_{M}(|\nabla^{2}\log
u|^{2}+Ric(L)(\nabla\log u,\nabla\log u))ud\mu.$ (34)

Proof. Based on Theorem 5.1, we can prove $(\ref{Ent-2})$ by the same argument
as used in the proof of Theorem 4.3 in [28]. Integrating by part yields

$\displaystyle\int_{M}{|Lu|^{2}\over
u}d\mu=-\int_{M}\left\langle\nabla\left({Lu\over u}\right),{\nabla u\over
u}\right\rangle ud\mu.$

Hence

$\displaystyle{d^{2}\over dt^{2}}H(u)$ $\displaystyle=$ $\displaystyle
2\int_{M}\left\langle\nabla Lu,{\nabla u\over u}\right\rangle
d\mu-\int_{M}Lu|\nabla\log u|^{2}d\mu$ $\displaystyle=$ $\displaystyle
2\int_{M}\left\langle\nabla Lu,{\nabla u\over u}\right\rangle
d\mu-\int_{M}uL|\nabla\log u|^{2}d\mu.$

By the Bochner formula for the Witten Laplacian, we have (see [28])

$\displaystyle L|\nabla\log u|^{2}=2\langle\nabla L\log u,\nabla\log
u\rangle+2|\nabla^{2}\log u|^{2}+2Ric(L)(\nabla\log u,\nabla\log u)$

Therefore

$\displaystyle{d^{2}\over dt^{2}}H(u)$ $\displaystyle=$ $\displaystyle
2\int_{M}\left\langle\nabla Lu,{\nabla u\over u}\right\rangle
d\mu-2\int_{M}\langle\nabla L\log u,\nabla u\rangle d\mu$ $\displaystyle\ \
-2\int_{M}(|\nabla^{2}\log u|^{2}+Ric(L)(\nabla\log u,\nabla\log u))ud\mu.$

Again, integrating by parts yields

$\displaystyle\int_{M}<\nabla L\log u,\nabla u>d\mu=-\int_{M}L\log u\cdot
Lud\mu=\int_{M}\langle\nabla\log u,\nabla Lu\rangle d\mu.$

Thus

$\displaystyle{d^{2}\over dt^{2}}H(u)=-2\int_{M}(|\nabla^{2}\log
u|^{2}+Ric(L)(\nabla\log u,\nabla\log u))ud\mu.$

The proof of Theorem 5.2 is completed. $\square$

###### Remark 5.3

Theorem 5.2 is an improvement of Theorem 4.3 in [28]. The entropy dissipation
formulas $(\ref{Ent-1})$ and $(\ref{Ent-2})$ play an important role in the
study of the convergence to the equilibrium measure of the heat equation
$\partial_{t}u=Lu$ and for the diffusion processes defining by the following
SDE on Riemannian manifold $(M,g)$:

$\displaystyle dX_{t}=dW_{t}-\nabla\phi(X_{t})dt,$ (35)

where $W_{t}$ is a Brownian motion on $(M,g)$. In the case where $M$ is
compact and $\phi\in C^{3}(M)$, it is well-known that the entropy dissipation
formulas $(\ref{Ent-1})$ and $(\ref{Ent-2})$ hold. However, as far as we know,
even in the non-weighted case, it seems that there is no precisely written
result in the literature (except our previous paper [28]) which ensures the
entropy dissipation formulas $(\ref{Ent-1})$ and $(\ref{Ent-2})$ on general
complete non-compact Riemannian manifolds $(M,g)$. Theorem 5.1 and Theorem 5.2
give us natural geometric conditions on $(M,g,\phi)$ to ensure $(\ref{Ent-1})$
and $(\ref{Ent-2})$.

Proof of Theorem 2.5. We simplify the proof of Theorem 2.3 in [28] as follows.
Let

$\displaystyle H_{m}(u,t)=-\int_{M}u\log ud\mu-{m\over 2}\left(1+\log(4\pi
t)\right).$

By Theorem 5.1 and Theorem 5.2, we have

$\displaystyle{d\over dt}H_{m}(u,t)=-\int_{M}uL\log ud\mu-{m\over
2t}=\int_{M}\left(|\nabla\log u|^{2}-{m\over 2t}\right)ud\mu,$ (36)

and

$\displaystyle{d^{2}\over dt^{2}}H_{m}(u,t)=-2\int_{M}(|\nabla^{2}\log
u|^{2}+Ric(L)(\nabla\log u,\nabla\log u))ud\mu+{m\over 2t^{2}}.$ (37)

By the definition formula $(\ref{W-H})$ of the $W$-entropy, we have

$\displaystyle W_{m}(u,t)={d\over dt}(tH_{m}(u,t))=H_{m}(u,t)+t{d\over
dt}H_{m}(u,t),$ (38)

and

$\displaystyle{d\over dt}W_{m}(u,t)=2{d\over dt}H_{m}(u,t)+t{d^{2}\over
dt^{2}}H_{m}(u,t).$ (39)

Substituting $(\ref{Ent-0})$ and $(\ref{Ent-4})$ into $(\ref{Ent-6})$, and
using $u={e^{-f}\over(4\pi t)^{m/2}}$, we obtain

$\displaystyle W_{m}(u,t)$ $\displaystyle=$ $\displaystyle-\int_{M}u\log
ud\mu-{m\over 2}\left(1+\log(4\pi t)\right)-t\int_{M}uL\log ud\mu-{m\over 2}$
$\displaystyle=$ $\displaystyle\int_{M}\left[t|\nabla\log u|^{2}-\log
u-m-{m\over 2}\log(4\pi t)\right]ud\mu$ $\displaystyle=$
$\displaystyle\int_{M}\left[t|\nabla f|^{2}+f-m\right]{e^{-f}\over(4\pi
t)^{m/2}}d\mu.$

This proves $(\ref{WW1})$ in Theorem 2.5. Substituting $(\ref{Ent-4})$ and
$(\ref{Ent-5})$ into $(\ref{Ent-7})$, we obtain

$\displaystyle{d\over dt}W_{m}(u,t)=-2\int_{M}uL\log
ud\mu-2\int_{M}t(|\nabla^{2}\log u|^{2}+Ric(L)(\nabla\log u,\nabla\log
u))ud\mu-{m\over 2t}.$

Noting that

$\displaystyle 2t|\nabla^{2}\log u|^{2}+{m\over 2t}=2t\left|\nabla^{2}\log
u-{g\over 2t}\right|^{2}+{m-n\over 2t}+2\Delta\log u,$

and by the integration by part formula

$\displaystyle\int_{M}\Delta\log uud\mu$ $\displaystyle=$
$\displaystyle\int_{M}(L\log u+\nabla\phi\cdot\nabla\log u)ud\mu$
$\displaystyle=$ $\displaystyle-\int_{M}|\nabla\log
u|^{2}ud\mu+\int_{M}\nabla\phi\cdot\nabla\log uud\mu,$

we have

$\displaystyle{d\over dt}W_{m}(u,t)$ $\displaystyle=$
$\displaystyle-2t\int_{M}\left|\nabla^{2}\log u-{g\over
2t}\right|^{2}ud\mu+2\int_{M}\nabla\phi\cdot\nabla\log uud\mu$
$\displaystyle-2\int_{M}tRic(L)(\nabla\log u,\nabla\log u)ud\mu-{m-n\over
2t}.$

By the same argument as in [28], the above identity yields $(\ref{WW2})$. The
proof of Theorem 2.5 is completed. $\square$

Proof of Theorem 2.6. The monotonicity part of Theorem 2.6 follows immediately
from Theorem 2.5. The rigidity part of Theorem 2.6 can be proved by the same
argument used in Theorem 2.4 in [28]. $\square$

##  6 LSI and Lower boundedness of the $W$-entropy

Following [41], we introduce the best logarithmic Sobolev constant

$\displaystyle\mu(\tau):=\inf\limits\left\\{W_{m}(u,\tau):\
u={e^{-f}\over(4\pi\tau)^{m/2}},\ \int_{M}ud\mu=1\right\\}.$

In [28], the author proved that $\mu(\tau)$ is always bounded from below on
compact Riemannian manifolds $M$ with $\phi\in C^{2}(M)$. In this section, we
prove a family of logarithmic Sobolev inequalities (LSI) on complete
Riemannian manifolds with $\mu$-bounded geometry condition.

Proof of Theorem 2.7. Let $v=\sqrt{u}$. Then we can rewrite $W_{m}(u,\tau)$ as
follows

$\displaystyle W_{m}(u,\tau)=4\tau\int_{M}|\nabla v|^{2}d\mu-\int_{M}v^{2}\log
v^{2}d\mu-\left(m+{m\over 2}\log(4\pi\tau)\right).$

We need to prove, for all $v\in C_{0}^{\infty}(M)$ with $\int_{M}v^{2}d\mu=1$,
we have

$\displaystyle 4\tau\int_{M}|\nabla v|^{2}d\mu-\int_{M}v^{2}\log
v^{2}d\mu-m\left(1+{\over 2}\log(4\pi\tau)\right)\geq\mu(\tau)>-\infty.$ (40)

Under the conditions of theorem, by Theorem 2.8 in [27], the following
$L^{2}$-Sobolev inequality holds: there exists a constant
$C_{1}(m,K,\mu_{0})>0$, depending only on $m,K,\mu_{0}$, such that

$\displaystyle\|f\|^{2}_{2m\over m-2}\leq C_{1}(m,K,\mu_{0})(\|\nabla
f||_{2}^{2}+\|f\|_{2}^{2}),\ \ \ \ \forall f\in C_{0}^{\infty}(M).$

By [12], the above $L^{2}$-Sobolev inequality implies that, for any
$\varepsilon>0$, there exists a constant $\beta(\varepsilon)>0$ such that the
following logarithmic Sobolev inequality holds

$\displaystyle\int_{M}f^{2}\log f^{2}d\mu\leq\varepsilon\int_{M}|\nabla
f|^{2}+\beta(\varepsilon)\|f\|_{2}^{2}+\|f\|_{2}^{2}\log\|f\|_{2}^{2},\ \ \ \
\forall f\in C_{0}^{\infty}(M),$ (41)

where for some constant $C_{2}(m,K,\mu_{0})>0$ depending only on
$m,K,\mu_{0}$, we have

$\beta(\varepsilon)\leq C_{2}(m,K,\mu_{0})-{m\over 2}\log\varepsilon.$

Applying $(\ref{logsob})$ to $f=v\in C_{0}^{\infty}(M)$ with
$\int_{M}v^{2}d\mu=1$, and taking $\varepsilon=4\tau$, we have

$\displaystyle\int_{M}v^{2}\log v^{2}d\mu\leq 4\tau\int_{M}|\nabla
v|^{2}-m\left(1+{1\over 2}\log(4\pi\tau)\right)-\mu(\tau),$

where

$-\mu(\tau):=\beta(4\tau)+m\left(1+{1\over 2}\log(4\pi\tau)\right),$

This proves $(\ref{www})$ with

$\displaystyle\mu(\tau)\geq-\left(C_{2}(m,K,\mu_{0})+m+{m\over
2}\log(4\pi)\right),\ \ \ \forall\tau>0.$

This finishes the proof of Theorem 2.7. $\square$

##  7 Analytic proof of Theorem 2.3

In this section, we use Hamilton’s Harnack inequality to give an analytic
proof of Theorem 2.3. This proof is similar to the one given in [22, 13] for
the first order gradient estimate of the logarithmic heat kernel of the
Laplace-Beltrami operator on complete Riemannian manifolds with non-negative
Ricci curvature.

Analytic proof of Theorem 2.3. Fix $T>0$, and let $u(t,x)$ be a positive and
bounded solution to the heat equation $\partial_{t}u=Lu$, $t\in(0,t_{1})$. Let

$A:=\sup\limits\\{u(t,x):0\leq t\leq t_{1},x\in M\\}.$

By Hamilton’s Harnack inequality $(\ref{E31b})$, we have

$\displaystyle t|\nabla\log u(t,x)|^{2}\leq(1+2Kt)\log(A/u(t,x)),\ \ \
\forall(t,x)\in[0,t_{1}]\times M.$ (42)

Let $s\in(0,T]$, $y\in M$, $t_{1}=s/2$ and $u(t,x)=p_{s/2+t}(x,y)$. By
$(\ref{EE1})$, $(\ref{heat-upp})$ and $(\ref{heat-lower})$, we have

$\displaystyle t|\nabla_{x}\log p_{s/2+t}(x,y)|^{2}\leq C_{K,m,T}(1+Kt)$
$\displaystyle\hskip
28.45274pt\times\left[1+{d(x,y)\over\sqrt{t}}+\log\left({C_{1}\over
C_{2}}{\mu(B(y,\sqrt{s/2+t}))\over\mu(B(y,\sqrt{s/2}))}\exp\left(C_{3}{d^{2}(x,y)\over
s/2+t}+C_{4}d(x,y)\right)\right)\right].$

In particular, taking $t=s/2$ and changing $s$ by $t$ we get

$\displaystyle{t\over 2}|\nabla_{x}\log p_{t}(x,y)|^{2}\leq
C_{K,m,T}(1+K{t\over 2})$ $\displaystyle\hskip
28.45274pt\times\left[1+{d(x,y)\over\sqrt{t}}+\log\left({C_{1}\over
C_{2}}{\mu(B(y,\sqrt{t}))\over\mu(B(y,\sqrt{t/2}))}\exp\left(C_{3}{d^{2}(x,y)\over
t}+C_{4}d(x,y)\right)\right)\right].$

By the Bishop-Gromov volume comparison $(\ref{E6})$, we derive

$\displaystyle t|\nabla_{x}\log p_{t}(x,y)|^{2}\leq
C_{K,m,T}\left(1+{d^{2}(x,y)\over t}+{d(x,y)\over\sqrt{t}}+d(x,y)\right),$

which yields

$\displaystyle|\nabla_{x}\log p_{t}(x,y)|\leq C_{K,m,T}\left({d(x,y)\over
t}+{1\over\sqrt{t}}\right).$

This finishes the analytic proof of Theorem 2.3. $\square$

##  8 Another probabilistic approach to the Hamilton type Harnack inequality

In this section we develop another probabilistic approach to the Hamilton type
Harnack inequality for the Witten Laplacian on complete Riemanian manifolds.

Let $X_{t}$ be the $L$-diffusion process on $M$, and let $U_{t}$ its
horizontal lift to $O(M)$. Let $\mathcal{F}_{t}=\sigma(X_{s},s\leq t)$ be the
natural complete $\sigma$-filed generated by $X_{t}$. Let $u$ be a positive
solution of the heat equation

$\displaystyle\partial_{t}u=L_{O(M)}u.$

Fix $T>0$. Let

$\displaystyle J(t,x)=\log u(T-t,\pi r),\ \ \ \forall r\in O(M),t\in[0,T].$

Now we introduce the following Doob $h$-transform for the $L$-diffusion
process $X_{t}$ on $M$. Let $\mathbb{P}_{x,T}$ be the law of the conditional
$L$-diffusion process starting from $x$ and with transition density

$\displaystyle\mathbb{E}_{x,T}(\xi)={1\over
u(T,x)}\mathbb{E}_{x}\left[u(T-t,X_{t})\xi\right],$

where $\xi$ is any bounded $\mathcal{F}_{t}$-measurable random variable. By
the Grirsanov transformation, we have

$\displaystyle\left.{d\mathbb{P}_{x,T}\over
d\mathbb{P}_{x}}\right|_{\mathcal{F}_{t}}={u(T-t,X_{t})\over
u(T,x)}=\exp\\{J(t,U_{t})-J(0,U_{0})\\},\ \ \ \forall t<T.$

By Itô’s formula, we have

$\displaystyle dJ(t,U_{t})={\partial J\over\partial
t}(t,U_{t})dt+\sqrt{2}\langle\nabla^{H}J(t,U_{t}),dB_{t}\rangle+L_{O(M)}J(t,U_{t})dt,$

Note that $J(t,u)$ satisfies the Hamilton-Jacobi equation

$\displaystyle{\partial J\over\partial t}+L_{O(M)}J+|\nabla^{H}J|^{2}=0,$ (43)

which yields

$\displaystyle\left.{d\mathbb{P}_{x,T}\over
d\mathbb{P}_{x}}\right|_{\mathcal{F}_{t}}=\exp\left[\int_{0}^{t}\langle\sqrt{2}\nabla^{H}J(s,U_{s}),dB_{s}\rangle-\int_{0}^{T}|\nabla^{H}J(s,U_{s})|^{2}ds\right],\
\ \ \forall t<T.$

It follows, by Girsanov’s theorem, that the process

$\displaystyle b_{t}=B_{t}-\sqrt{2}\int_{0}^{t}\nabla^{H}J(s,U_{s})ds,\ \ \
s\in[0,T),$ (44)

is a Brownian motion under $\mathbb{P}_{x,T}$. Substituting $(\ref{b1})$ into
$(\ref{U})$ and $(\ref{X})$, we have the following lemma.

###### Lemma 8.1

The conditional $L$-diffusion process $X_{t}$ on $M$ staring from $x$ with
transition semigroup $\mathbb{P}_{x,T}$, and its horizontal lift $U_{t}$, are
the solutions of the following Stratonovich SDEs:

$\displaystyle dU_{t}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}H_{i}(U_{t})\circ\left[\sqrt{2}db_{t}^{i}+2H_{i}\log
u(T-t,U_{t})dt\right]-\nabla^{H}\phi(U_{t})dt,$ $\displaystyle dX_{t}$
$\displaystyle=$ $\displaystyle U_{t}\circ\left[\sqrt{2}db_{t}+2\nabla^{H}\log
u(T-t,U_{t})dt\right]-\nabla\phi(X_{t})dt,$

where $b_{t}$ is a standard Brownian motion on $\mathbb{R}^{n}$ under
$\mathbb{P}_{x,T}$.

###### Proposition 8.2

For all $0<t<T$, we have

$\displaystyle\mathbb{E}[d|\nabla^{H}J(t,U_{t})|^{2}]=2\mathbb{E}\left[|\nabla^{H}\nabla^{H}J|^{2}+(Ric+\nabla^{2}\phi)(\nabla^{H}J,\nabla^{H}J)\right].$
(45)

Proof. The proof is as the same as the one of Proposition 4.2. $\square$

###### Theorem 8.3

Suppose that $Ric+\nabla^{2}\phi\geq-K$. Then for all $T>0$, we have

$\displaystyle|\nabla\log u(T,x)|^{2}\leq 2\left({1\over
T}+K\right)\mathbb{E}\left[\log{u(T/2,X_{T/2})\over u(T,x)}\right].$ (46)

Proof. The proof is as the same as the one of Proposition 4.3. $\square$

As a corollary, we can derive the following Hamilton type Harnack inequality
for the Witten Laplacian on complete Riemanian manifolds.

###### Corollary 8.4

Let $u$ be a bounded and positive solution of the heat equation
$\partial_{t}u=Lu$. Suppose that $Ric+\nabla^{2}\phi\geq-K$. Then for all
$T>0$, we have

$\displaystyle|\nabla\log u(T,x)|^{2}\leq 2\left({1\over
T}+K\right)\log\left({A/u(T,x)}\right),$ (47)

where

$\displaystyle A:=\sup\limits\\{u(t,x):x\in M,t\in[0,T]\\}.$

Proof. The Hamilton type Harnack inequality $(\ref{E3bc})$ follows from
$(\ref{E3a})$ in Theorem 8.3. $\square$

Acknowledgement. The author would like to thank Fabrice Baudoin for helpful
discussion on the alternative proof of Theorem 2.1 in September 2011. Part of
this work was done when the author visited Rutgers University and Purdue
University in April-May 2012. The author would like to thank Eric Carlen and
Fabrice Baudoin for their invitations and very nice hospitalities. He would
like to thank E. Hsu and Z.-Q. Lu for the discussions and for their interests
on this work. The revised version is done when the author visits the Institut
des Hautes Etudes Scientifiques in April 1-June 1, 2013. He would like to
express his gratitude to J.-M. Bismut and J.-P. Bourguignon for their kind
helps and invitation which allow him to finish this work in a very nice
academic environment in IHES.

## References

  * [1] M. Arnaudon, A. Thalmaier, Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis, Advanced Studies in Pure Mathematics, 57 (2010) 29-48.
  * [2] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Gradient estimate and Harnack inequality on non-compact Riemannian manifolds, Stochastic Processes Appl. 119 (2009) 3653-3670.
  * [3] D. Bakry, Un critère de non-explosion pour certaines diffusions sur une variété riemannienne complète, C. R. Acad. Sci. Paris, Sér. I 303 (1986) 23-26.
  * [4] D. Bakry, Functional inequalities for Markov semigroups, Notes de cours du Tata Institute, Bombay, 2002. In Probability measures on groups: recent directions and trends, 91-147. Proceedings of the CIMPA-TIFR School (2002), Tata Institute of Fundamental Research. Narosha Publishing House, 2006.
  * [5] D. Bakry, M. Emery, Diffusion hypercontractives, Sém. Prob. XIX, Lect. Notes in Maths. 1123 (1985), 177-206.
  * [6] D. Bakry, M. Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality, Rev. Mat. Iberoam. 22 (2006), No. 2, 683-702.
  * [7] K. Brighton, A Liouville-type theorem for smooth metric measure spaces, arXiv:1006.0751
  * [8] H.-D. Cao, X.-P. Zhu, A complete proof of the Poincaré and geometrization conjectures: application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math. 10 (2006), no. 2, 165-492, and Asian J. Math. 10 (2006), no. 4, 663.
  * [9] S.-C. Chang, T.-J. Kuo, S.-H. Lai, Li-Yau gradient estimate and entropy formulae for the CR heat equation in a closed pseudohermitian 3-manifold, J. Diff. Geom. 89, No. 2 (2011), 185-216.
  * [10] B. Chow, P. Lu, L. Ni, Hamilton’s Ricci Flow, Lectures in Comtemporary Maths., Sciences Press, Beijing, Amer. Math. Soc. 2006\.
  * [11] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, P. Lu, L. Ni, The Ricci Flow, Part I: Techniques and Applications-Geometric Aspects, Mathematical Surveys and Monographs Ser. 2007.
  * [12] E.B. Davies, Heat Kernel and spectral Theory, Cambridge University Press, Cambridge, New York, 1990.
  * [13] B. Driver, M. Gordia, Integrated Harnack inequalities on Lie groups, J. Diff. Geom. 83 (2009) 501-550
  * [14] K. Ecker, A formula relating entropy monotonicity to Harnack inequalities, Comm. Anal. Geom. 15 (2007), no. 5, 1025–1061.
  * [15] A. Engoulatov, A universal bound on the gradient of logarithm of the heat kernel for manifolds with bounded Ricci curvature. J. Funct. Anal. 238, 518-529 (2006)
  * [16] F. Fang, X.-D. Li, Z.-L. Zhang, Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 2, 563-573.
  * [17] A. Futaki, H.-Z. Li, X.-D. Li, On the first eigenvalue of the Witten Laplacian and the diameter of compact shrinking Ricci solitons, arxiv.org/abs/1111.6364v3
  * [18] R.S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993), no. 1, 113-126.
  * [19] R.S. Hamilton, The formation of singularities in the Ricci flow, Surveys in Differential Geometry, 2, 7-136, International Press, 1995.
  * [20] E. Hsu, Estimates of derivatives of the heat kernel on a compact Riemannian manifold. Proc. Am. Math. Soc. 127 (1999), 3739 3744.
  * [21] B. Kleiner, J. Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), no. 5, 2587-2855.
  * [22] B. Kotschwar, Hamilton s gradient estimate for the heat kernel on complete manifolds, Proc. of Amer. Math. Soc. 135 (2007), no. 9, 3013-3019.
  * [23] B. Kotschwar, L. Ni, Gradient estimate for $p$-harmonic functions, $1/H$ flow and an entropy formula, Ann. Sci. Ec. Norm. Sup., Vol 42 (2009), no. 1, 1–36.
  * [24] P. Li, S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta. Math. 156 (1986), 153-201.
  * [25] X.-D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl. 84 (2005), 1295-1361.
  * [26] X.-D. Li, On the Strong Liuville theorem for weighted Laplacian on complete Riemannian manifolds, preprint, 2007.
  * [27] X.-D. Li, Sobolev inequalities on forms and $L^{p,q}$-cohomology on complete Riemannian manifolds, J. Geom. Anal. 20 (2010), 354-387.
  * [28] X.-D. Li, Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature, Math. Ann. 353 (2012), 403-437.
  * [29] X.-D. Li, Perelman’s W-entropy for the Fokker-Planck equation over complete Riemannian manifolds, Bull. Sci. Math. 135 (2011) 871-882.
  * [30] S. Li, X.-D. Li, Perelman s entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials, arXiv:1303.6019v1
  * [31] P. Lu, L. Ni, J.-J. Vazquez, C. Villani, Local Aronson-Benilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds, Jour. Math. Pures Appl., Vol 91 (2009), no. 1, 1-19.
  * [32] J. Lott, Some geometric properties of the Bakry-Emery Ricci tensor. Comment. Math. Helv. 78 (2003), 865-883.
  * [33] P. Malliavin, Formule de la moyenne, calcul des perturbations et th orie d annulation pour les formes harmoniques, J. Funct. Anal. 17 (1974), 274-291.
  * [34] P. Malliavin, Stochastic Analysis. Springer, Berlin, 1997.
  * [35] J. W. Morgan, G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs, 3. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007\.
  * [36] O. Munteanu, J. Wang, Smooth metric measure spaces with non-negative curvature, arxiv.1103.0746v2.
  * [37] O. Munteanu, J. Wang, Analysis of weightd Laplacian and applications to Ricci solitons, arxiv.org/abs/1112.3027v1
  * [38] Z.-M. Qian, Estimates for weight volumes and applications, Quart. J. Math. Oxford Ser. 48 (1987), 235-242.
  * [39] L. Ni, The entropy formula for linear equation, J. Geom. Anal. 14 (1), 87-100, (2004).
  * [40] L. Ni, Addenda to “The entropy formula for linear equation”, J. Geom. Anal. 14 (2), 329-334, (2004).
  * [41] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, http://arXiv.org/abs/maths0211159.
  * [42] S.-J. Sheu, Some estimates of the transition density of a nondegenerate diffusion Markov process. Ann. Probab. 19(2), 538-561 (1991)
  * [43] P. Souplet, Q. S. Zhang, Sharp gradient estimate and Yau s Liouville theorem for the heat equation on noncompact manifolds. Bull London Math Soc, 38 (2006), 1045-1053.
  * [44] D. Stroock, J. Turesky, Upper bounds on derivatives of the logarithm of the heat kernel. Commun. Anal. Geom. 6, 669 685 (1998).
  * [45] R. Schoen, S.-T. Yau, Lectures on Differential Geometry, International Press, 1994.
  * [46] S.T. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J., (1976), 659-670.
  * [47] C. Villani, Optimal Transport, Old and New, Springer, 2008.
  * [48] F.-Y. Wang, Equivalent semigroup properties for curvature-dimension condition, arxiv.org/abs/1012.5686v1
  * [49] G. Wei, W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Diff. Geom. (2009), 377-405.