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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 Lev́y 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 ℝd with smooth variable index α(x) ∈ (0, 2) a transience criterion in terms of the exponent α(x); if d =1 and infx∈ℝα(x) ∈ (1, 2), then the stable-like process has local times. ©2012 American Mathematical Society.

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