Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms

TL;DR: In this article , the sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain are studied. But the authors focus on a special case of the processes under consideration.

TL;DR: In this paper , the authors presented an off-diagonal upper estimate of the heat kernel for any regular Dirichlet form without a killing part on the doubling space, and obtained the weighted L2{L^{2}}-norm estimate of survival function 1-PtB Ω(1B{1-P_{t}^{B}1_{B}) for any metric ball B.

TL;DR: In this paper, it was shown that the parabolic Harnack inequality implies the existence of a jump kernel for symmetric pure jump processes, and the key ingredients of their proof are the Levy system formula and estimates on the heat kernel.

TL;DR: In this article, the authors prove sharp estimates on heat kernels and Green functions for subordinate Markov processes with both discrete an continuous time, under relatively weak assumptions about original processes as well as Laplace exponents of subordinators.

TL;DR: In this paper, the authors considered symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and established stability of two-sided heat kernel estimates and heat kernel upper bounds.

TL;DR: In this paper, the transition densities of pure-jump symmetric Markov processes were studied under mild assumptions on their scale functions, and two-sided estimates of the heat kernel estimates for such processes were established.

TL;DR: In this article, the authors studied the existence and uniqueness of strong and weak solutions for general time fractional Poisson equations and showed that there is an integral representation of the solutions with zero initial values in terms of semigroup for the infinitesimal spatial generator L and the corresponding subordinator associated with the time-fractional derivative.

TL;DR: In this paper, the authors studied the existence and uniqueness of strong and weak solutions for general time fractional Poisson equations and showed that there is an integral representation of the solutions with zero initial values in terms of semigroup for the infinitesimal spatial generator and the corresponding subordinator associated with the time-fractional derivative.