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Schauder Estimates for Equations Associated with Lévy Generators

  • Franziska KühnEmail author
Article

Abstract

We study the regularity of solutions to the integro-differential equation \(Af-\lambda f=g\) associated with the infinitesimal generator A of a Lévy process. We show that gradient estimates for the transition density can be used to derive Schauder estimates for f. Our main result allows us to establish Schauder estimates for a wide class of Lévy generators, including generators of stable Lévy processes and subordinated Brownian motions. Moreover, we obtain new insights on the (domain of the) infinitesimal generator of a Lévy process whose characteristic exponent \(\psi \) satisfies \(\text {Re} \, \psi (\xi ) \asymp |\xi |^{\alpha }\) for large \(|\xi |\). We discuss the optimality of our results by studying in detail the domain of the infinitesimal generator of the Cauchy process.

Keywords

Lévy process Integro-differential equation Schauder estimate Hölder space Gradient estimate 

Mathematics Subject Classification

Primary 60G51 Secondary 45K05 60J35 

Notes

Acknowledgements

I am grateful to Niels Jacob and René Schilling for valuable comments which helped to improve the presentation of this paper; I owe the proof of Remark 3.2(ii) to René Schilling. Moreover, I thank the Institut national des sciences appliquées de Toulouse, Génie mathématique et modélisation for its hospitality during my stay in Toulouse, where a part of this work was accomplished.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseUniversité Paul Sabatier III ToulouseToulouseFrance

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