Upper Rate Functions of Brownian Motion Type for Symmetric Jump Processes

Let $X$ be a symmetric jump process on ${\mathbb{R}}^d$ such that the corresponding jumping kernel $J(x,y)$ satisfies $$J(x,y)\le \frac{c}{\mathrm{\mid }x-y{\mathrm{\mid }}^{d+2}{\log }^{1+\epsilon }(e+\mathrm{\mid }x-y\mathrm{\mid })}$$ for all $x,y\in {\mathbb{R}}^d$ with $\mathrm{\mid }x-y\mathrm{\mid }\ge 1$ and some constants $c,\epsilon >0$. Under additional mild assumptions on $J(x,y)$ for $\mathrm{\mid }x-y\mathrm{\mid }<1$, 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 $$\frac{1}{\mathrm{\mid }x-y{\mathrm{\mid }}^{d+2+\epsilon }}$$ for all $x,y\in {\mathbb{R}}^d$ with $\mathrm{\mid }x-y\mathrm{\mid }\ge 1$ and some constant $\epsilon >0$.