Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part

TL;DR

This paper establishes heat kernel estimates and Harnack inequalities for a class of Dirichlet forms combining local and non-local parts under ellipticity and kernel conditions.

Contribution

It provides new upper and lower bounds for heat kernels and proves Harnack inequalities for non-local Dirichlet forms with elliptic local parts.

Findings

Derived heat kernel bounds under specified conditions.

Proved Harnack inequality for harmonic functions.

Established regularity results for solutions.

Abstract

We consider the Dirichlet form given by \sE(f,f)&=&{1/2}\int_{\bR^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx &+&\int_{\bR^d\times \bR^d} (f(y)-f(x))^2J(x,y)dxdy. Under the assumption that the $\{a_{ij}\}$ are symmetric and uniformly elliptic and with suitable conditions on $J$, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to $\sE$.