Abstract

We study the domain of the generator of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder–Zygmund spaces are in the domain. We use toolsfrom probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process, $$                 \lim_{t\to 0} (E^xf(X_t)-f(x))/t $$ for functions $f$ which are not necessarily bounded or differentiable. The pointwise limit exists for fixed if $f$ satisfies a Hölder condition at x. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes.