On the domain of fractional Laplacians and related generators of Feller processes |
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Authors: | Franziska Kühn René L. Schilling |
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Affiliation: | 1. Institut de Mathématiques de Toulouse, Université Paul Sabatier III Toulouse, 118 Route de Narbonne, 31062 Toulouse, France;2. TU Dresden, Fakultät Mathematik, Institut für Mathematische Stochastik, 01062 Dresden, Germany |
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Abstract: | In this paper 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 tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process , 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. As an application we prove that the domain of the generator of contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes. |
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Keywords: | primary 60J75 secondary 60G51 60J25 60J35 Lévy-type processes Blumenthal–Getoor index Small-time asymptotics Hölder space of variable order |
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