LOWER INEQUALITIES OF HEAT SEMIGROUPS BY USING PARABOLIC MAXIMUM PRINCIPLE

Using parabolic maximum principle, we apply the analytic method to obtain lower comparison inequalities for non-negative weak supersolutions of the heat equation associated with a regular strongly p-local Dirichlet form on the abstract metric measure space. As an application we obtain lower estimates for heat kernels on some Riemannian manifolds.

热核下界估计的一类推广

主要利用分析、概率和随机过程的方法,研究热核下界估计的条件和估计式.用γ-链条件作为中点性质的推广,得到了局部紧度量空间上热核下界估计更容易满足和验证的条件以及热核下界估计式.

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.

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.

Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradients

Using Duhamel’s formula,we prove sharp two-sided estimates for the spectral fractional Laplacian’s heat kernel with time-dependent gradient perturbation in bounded C^1,1 domains.In addition,we obtain a gradient estimate as well as the Holder continuity of the heat kernel’s gradient.

On-diagonal Heat Kernel Estimates for Schrödinger Semigroups and Their Application

We establish explicit and sharp on-diagonal heat kernel estimates for Schrödinger semigroups with unbounded potentials corresponding to a large class of symmetric jump processes.The approach is based on recent developments on the two-sided(Dirichlet)heat kernel estimates and intrinsic contractivity properties for symmetric jump processes.As a consequence,we present a more direct argument to yield asymptotic behaviors for eigenvalues of associated nonlocal operators.

退化椭圆算子的特征值问题

本文简要介绍退化椭圆算子的特征值问题的研究结果与研究方法;以有限阶退化椭圆算子为主线,主要阐述研究其Dirichlet特征值上下界估计和渐近估计的方法与结论.