Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators

Franziska Kühn ()
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Franziska Kühn: TU Dresden

Journal of Theoretical Probability, 2021, vol. 34, issue 3, 1506-1578

Abstract: Abstract We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation $$Af=g$$ A f = g associated with the (extended) infinitesimal generator of a Feller process. The regularity of f is described in terms of Hölder–Zygmund spaces of variable order and, moreover, we establish Schauder estimates. Since Hölder estimates for Feller semigroups have been intensively studied in the last years, our results apply to a wide class of Feller processes, e.g. random time changes of Lévy processes and solutions to Lévy-driven stochastic differential equations. Most prominently, we establish Schauder estimates for the Poisson equation associated with the fractional Laplacian of variable order. As a by-product, we obtain new regularity estimates for semigroups associated with stable-like processes.

Keywords: Feller process; Infinitesimal generator; Regularity; Hölder space of variable order; Favard space; Primary 60J25; Secondary 45K05; 35B65; 60J35; 60J75 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-020-01008-x

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