Upper Rate Functions of Brownian Motion Type for Symmetric Jump Processes

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 $\varepsilon>0$.