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 $B(0,R_0)\subseteq \mathrm{supp}(\nu )$ for some constant $R_0>0$, where $\mathrm{supp}(\nu )$ denotes the support of the associated Lévy measure $\nu$. 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.

References in corpus (3)

• Estimates of transition densities and their derivatives for jump Lévy processes
• Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of Lévy processes
• Intrinsic Ultracontractivity of Feynman-Kac Semigroups for Symmetric Jump Processes