On Heat Kernel Estimates and Parabolic Harnack Inequality for Jump Processes on Metric Measure Spaces

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.

SOBOLEV INEQUALITY ON RIEMANNIAN MANIFOLDS

Let M be an n dimensional complete Riemannian manifold satisfying the doublingvolume property and an on-diagonal heat kernel estimate. The necessary-sufficientcondition for the Sobolev inequality ‖f‖q ≤ Cn,,v,p,q(‖▽f‖p+‖fp) (2≤p＜q＜∞) is given.

DIFFUSION PROCESSES ON COMPLETE RIEMANNIAN MANIFOLDS

In this paper,a basic estimate for the conditional Riemannian Brownian motion on a complete manifold with non-negative Ricci curvature is established.Applying it to the heat kernel estimate of the operator 1/2△+b,we obtain the Aronson′s estimate for the operator 1/2△+b,which can be regarded as an extension of Peter Li and S.T.Yau's heat kernel estimate for the Laplace-Beltrami operator.

Symmetric jump processes and their heat kernel estimates

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently,a class of symmetric integro-differential operators).We focus on the sharp two-sided estimates for the transition density functions(or heat kernels) of the processes,a priori Hlder estimate and parabolic Harnack inequalities for their parabolic functions.In contrast to the second order elliptic differential operator case,the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.

带边紧流形上热核的Harnack不等式

198 6年 ,P.Li与丘成桐给出了带凸边界的紧黎曼流形上关于热核的一个 Harnack不等式 (可参看 [6]) 。

Li-Yau抛物不等式的局部指数形式

证明了Li-Yau抛物不等式局部指数形式，它是具有负下界Ricci曲率的完备流形上热方程正解的一种新的梯度估计．作为它的应用，可以得到热方程解的局部Hamack估计和热核的Gauss下界估计．

ON THE UPPER ESTIMATES OF FUNDAMENTAL SOLUTIONS OF PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS

The authors first derive the gradient estimates and Harnack inequalties for positive solutions of the equation on(x, t)An(x, t) + b(x, t)' ac(￣, t) + h(x, t)u(x, t) ￣ m ￣ 0on complete Riemannian manifolds, and then derive the upper bounds of any positive L2fundamental solution of the equation when h(x, t) and b(x, t) are independent of t.

黎曼流形上的一般Sobolev型不等式

给出了流形上与一般Sobolev型不等式等价的等周不等式和热核上界条件 。

紧致黎曼流形上对数热核的高阶导数估计

设M为一个d-维紧致黎曼流形,对任意的t∈(0,1],x,y∈M,记pM(t,x,y)是M的极小热核.本文利用流形M上的水平布朗桥,把文献[1]中关于对数热核lnpM(t,x,y)的单变量的高阶导数估计推广到关于(x,y)两个变量上,即对于任意的非负整数n,m,都存在依赖于n,m和流形M的常数C使得下式成立:|▽_x^n▽_y^mlnpM(t,x,y)|≤C[d(x,y)/t+1/t^(1/2)]^(n+m).

紧Riemann流形的高阶特征值估计

对Ricci曲率具负下界的紧Riemann流形,本文获得了热方程正解优化的梯度估计及Harnack不等式,证明了高阶特征值下界定量估计的猜想．