Some Theorems on Feller Processes: Transience, Local Times and Ultracontractivity

TL;DR

This paper establishes conditions for transience, local times, and ultracontractivity of Feller processes, with sharp results for Lévy processes, using symmetrization and characteristic function bounds.

Contribution

It provides new sufficient conditions for key properties of Feller processes, including transience and local times, and introduces a transience criterion for stable-like processes with variable index.

Findings

Conditions for transience and local times of Feller processes.

Sharp criteria for Lévy processes.

Transience criterion for stable-like processes with variable index.

Abstract

We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for L\'{e}vy processes. The proof uses a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process. As a byproduct, we obtain for stable-like processes (in the sense of R.\ Bass) on $\R^d$ with smooth variable index $\alpha(x)\in(0,2)$ a transience criterion in terms of the exponent $\alpha(x)$; if $d=1$ and $\inf_{x\in\R} \alpha(x)\in (1,2)$, then the stable-like process has local times.