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

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 Lévy 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 ${\mathbb{R}}^d$ with smooth variable index $\alpha (x)\in (0,2)$ a transience criterion in terms of the exponent $\alpha (x)$; if $d=1$ and ${\inf }_{x\in \mathbb{R}}\alpha (x)\in (1,2)$, then the stable-like process has local times.

arXiv comment34 pages