Heat kernel estimates for symmetric Markov processes in $C^{1,\eta}$ open sets

Abstract

In this thesis, we establish sharp two-sided heat kernel estimates for a large class of symmetric Markov processes in $C^{1,\eta}$ 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}

^{-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 $c_1e^{c_2r^{\beta}}\le \psi(r)\le c_3e^{c_4r^{\beta}}$ 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

^{\rho}$ for $

<1$ and $\rho>\alpha/2$. As a corollary of our main result, we estimates sharp two-sided Green function for this process in $C^{1,\eta}$ open sets.