Global heat kernel estimates for symmetric Markov processes dominated by stable-like processes in exterior $C^{1,\eta }$ open sets

Abstract

In this paper, we establish sharp two-sided heat kernel estimates for a large class of symmetric Markov processes in exterior $C^{1,\eta }$ open sets for all $t>0$. The processes are symmetric pure jump Markov processes with jumping kernel intensity $$\kappa (x,y)\psi (|x-y|{)}^{-1}|x-y{|}^{-d-\alpha }$$ where $\alpha \in (0,2)$, $\psi$ is an increasing function on $[0,\infty )$ with $\psi (r)=1$ on $0<r\le 1$ and $c_1{e}^{c_2{r}^{\beta }}\le \psi (r)\le c_3{e}^{c_4{r}^{\beta }}$ on $r>1$ for $\beta \in [0,\infty ]$. A symmetric function $\kappa (x,y)$ is bounded by two positive constants and $|\kappa (x,y)-\kappa (x,x)|\le c_5|x-y{|}^{\rho }$ for $|x-y|<1$ and $\rho >\alpha /2$. As a corollary of our main result, we estimates sharp two-sided Green function for this process in $C^{1,\eta }$ exterior open sets.

arXiv admin note: text overlap with arXiv:1112.2778 by other authors