Non-local Dirichlet Forms and Symmetric Jump Processes

TL;DR

This paper studies symmetric non-local Dirichlet forms with position-dependent jump intensities, establishing heat kernel estimates, constructing associated Markov processes, proving a parabolic Harnack inequality, and providing an example of discontinuous harmonic functions.

Contribution

It introduces a framework for non-local Dirichlet forms with variable jump kernels, extending analysis of jump processes and harmonic functions.

Findings

Established upper and lower heat kernel estimates.

Constructed a strong Markov process associated with the form.

Proved a parabolic Harnack inequality for solutions.

Abstract

We consider the symmetric non-local Dirichlet form $(E, F)$ given by \[ E (f,f)=\int_{R^d} \int_{R^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $F$ the closure of the set of $C^1$ functions on $R^d$ with compact support with respect to $E_1$, where $E_1 (f, f):=E (f, f)+\int_{R^d} f(x)^2 dx$, and where the jump kernel $J$ satisfies \[ \kappa_1|y-x|^{-d-\alpha} \leq J(x,y) \leq \kappa_2|y-x|^{-d-\beta} \] for $0<\alpha< \beta <2, |x-y|<1$. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to $(E, F)$. We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to $E$. Finally we construct an example where the corresponding harmonic functions need not be continuous.