Some theorems on Feller processes: Transience, local times and ultracontractivity

Schilling, René; Wang, Jian

doi:10.1090/S0002-9947-2012-05738-2

Some theorems on Feller processes: Transience, local times and ultracontractivity
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by René L. Schilling and Jian Wang PDF

Trans. Amer. Math. Soc. 365 (2013), 3255-3286 Request permission

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 by-product, 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.

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