Some theorems on Feller processes: transience, local times and ultracontractivity (Q2838122)

The authors prove ``sufficient conditions for the transience and the existence of local times of a Feller process''. Furthermore, they establish ``the ultracontractivity of the associated Feller semigroup''. Their conditions turn out to be sharp for Lévy processes. As they point out, their main tool is the fact that the generator of the semigroup is a pseudodifferential operator with continuous negative definite symbol. Their proofs use ``a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process''. As an example the authors consider \(d\)-dimensional stable-like processes in the sense of R. Bass. For this class of processes they prove a transience criterion in terms of the exponent \(\alpha(x)\) if the variable index \(\alpha(x)\in (0,2)\) is smooth; in dimension one the stable-like process has local times if \(\inf_{x\in\mathbb{R}}\in(1,2)\). Some ``necessary properties and estimates [...] are proved in a simple and self-contained way in the appendix''.