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

TL;DR

This paper establishes intrinsic ultracontractivity for a broad class of Le9vy processes killed upon exiting bounded open sets, under conditions on the support of their Le9vy measure, with weaker assumptions for symmetric cases.

Contribution

It extends the understanding of ultracontractivity to general (not necessarily symmetric) Le9vy processes in bounded domains without boundary regularity requirements.

Findings

Proves intrinsic ultracontractivity for non-symmetric Le9vy processes with support conditions.

Establishes ultracontractivity for symmetric Le9vy processes in Hf6lder domains under weaker assumptions.

Provides conditions on the Le9vy measure support ensuring ultracontractivity.

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.