Upper Rate Functions of Brownian Motion Type for Symmetric Jump Processes

TL;DR

This paper establishes that symmetric jump processes with specific jump kernel bounds have an upper rate function similar to Brownian motion, using heat kernel estimates for large time behavior.

Contribution

It provides the first upper rate function for symmetric jump processes with kernels decaying like rac{1}{|x-y|^{d+2} ext{log}^{1+ ext{ε}}(e+|x-y|)}, extending Brownian motion results.

Findings

Upper rate function of order \, ext{C}\sqrt{r ext{log} ext{log} r} for the process.

Two-sided heat kernel estimates for jump processes with kernels comparable to rac{1}{|x-y|^{d+2+ ext{ε}}}.

Method based on large-time heat kernel estimates.

Abstract

Let $X$ be a symmetric jump process on $\R^d$ such that the corresponding jumping kernel $J(x,y)$ satisfies $$J(x,y)\le \frac{c}{|x-y|^{d+2}\log^{1+\varepsilon}(e+|x-y|)}$$ for all $x,y\in\R^d$ with $|x-y|\ge1$ and some constants $c,\varepsilon>0$. Under additional mild assumptions on $J(x,y)$ for $|x-y|<1$, we show that $C\sqrt{r\log \log r}$ with some constant $C>0$ is an upper rate function of the process $X$, which enjoys the same form as that for Brownian motions. The approach is based on heat kernel estimates of large time for the process $X$. As a by-product, we also obtain two-sided heat kernel estimates of large time for symmetric jump processes whose jumping kernels are comparable to $$\frac{1}{|x-y|^{d+2+\varepsilon}}$$ for all $x,y\in\R^d$ with $|x-y|\ge1$ and some constant $\varepsilon>0$.