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 by-product, we obtain for stable-like processes (in the sense of R. Bass) on R d \mathbb {R}^d with smooth variable index α ( x ) ∈ ( 0 , 2 ) \alpha (x)\in (0,2) a transience criterion in terms of the exponent α ( x ) \alpha (x) ; if d = 1 d=1 and inf x ∈ R α ( x ) ∈ ( 1 , 2 ) \inf _{x\in \mathbb {R}} \alpha (x)\in (1,2) , then the stable-like process has local times.