Intrinsic Ultracontractivity for General Lévy processes on Bounded Open Sets

Abstract

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