Heat kernel estimates and their stabilities for symmetric jump processes with general mixed polynomial growths on metric measure spaces

TL;DR

This paper derives stable heat kernel estimates for symmetric jump processes on metric measure spaces with mixed polynomial growth, extending previous results to more general jump behaviors and connecting these estimates to probabilistic laws.

Contribution

It establishes new stability results for heat kernel estimates under general mixed polynomial growth conditions and relates these to moment conditions and laws of iterated logarithm.

Findings

Heat kernel estimates are stable under general mixed polynomial growth.

Extension of heat kernel bounds to spaces with sub-Gaussian diffusion.

Equivalence between moment conditions and laws of iterated logarithm.

Abstract

In this paper, we consider a symmetric pure jump Markov process $X$ on a metric measure space with volume doubling conditions. Our focus is on estimating the transition density $p(t,x,y)$ of $X$ and studying its stability when the jumping kernel exhibits general mixed polynomial growth. Unlike previous work, in our setting, the rate function governing the jump growth may not be comparable to the scale function that determines whether $p(t,x,y)$ has near-diagonal or off-diagonal estimates. Under the assumption that lower scaling index of scale function is greater than $1$, we establish stabilities of heat kernel estimates. Additionally, if the metric measure space admits a conservative diffusion process with a transition density satisfying sub-Gaussian bounds, we generalize heat kernel estimates from [3, Theorems 1.2 and 1.4] using the rate function and the function $F$ related to walk dimension of underlying space. As an application, we prove the equivalence between a finite moment condition based on $F$ and a generalized Khintchine-type law of iterated logarithm at infinity for symmetric Markov processes.