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

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  • Franziska Kühn

    (TU Dresden)

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.

Suggested Citation

  • Franziska Kühn, 2021. "Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1506-1578, September.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:3:d:10.1007_s10959-020-01008-x
    DOI: 10.1007/s10959-020-01008-x
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    References listed on IDEAS

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    1. Luo, Dejun & Wang, Jian, 2019. "Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3129-3173.
    2. Kulik, Alexei M., 2019. "On weak uniqueness and distributional properties of a solution to an SDE with α-stable noise," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 473-506.
    3. Mingjie Liang & Jian Wang, 2019. "Spatial regularity of semigroups generated by Lévy type operators," Mathematische Nachrichten, Wiley Blackwell, vol. 292(7), pages 1551-1566, July.
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