Censored symmetric L\'evy processes

TL;DR

This paper explores three equivalent ways to construct censored symmetric discontinuous Lévy processes on open sets, analyzes their boundary behavior using advanced functional analysis, and establishes key inequalities like the Harnack inequality.

Contribution

It introduces a unified framework for constructing censored Lévy processes and extends boundary analysis and inequalities to these processes under new conditions.

Findings

Equivalent constructions of censored Lévy processes are established.

Boundary behavior is characterized using Besov-type space analysis.

Scale-invariant Harnack inequality is proved for the censored processes.

Abstract

We examine three equivalent constructions of a censored symmetric purely discontinuous L\'evy process on an open set $D$; via the corresponding Dirichlet form, through the Feynman-Kac transform of the L\'evy process killed outside of $D$ and from the same killed process by the Ikeda-Nagasawa-Watanabe piecing together procedure. By applying the trace theorem on $n$-sets for Besov-type spaces of generalized smoothness associated with complete Bernstein functions satisfying certain scaling conditions, we analyze the boundary behaviour of the corresponding censored L\'evy process and determine conditions under which the process approaches the boundary $\partial D$ in finite time. Furthermore, we prove a stronger version of the 3G inequality and its generalized version for Green functions of purely discontinuous L\'evy processes on $\kappa$-fat open sets. Using this result, we obtain the scale invariant Harnack inequality for the corresponding censored process.