Intrinsic Ultracontractivity for General L\'evy processes on Bounded Open Sets

Abstract

We prove that a general (not necessarily symmetric) L\'evy process killed on exiting a bounded open set (without regular condition on the boundary) is intrinsically ultracontractive, provided that $B(0,R_0)\subseteq \rm{supp}(\nu)$ for some constant $R_0>0$, where $\rm{supp}(\nu)$ denotes the support of the associated L\'evy measure $\nu$. For a symmetric L\'evy process killed on exiting a bounded H\"older domain of order $0$, we also obtain the intrinsic ultracontractivity under much weaker assumption on the associated L\'evy measure.