L1 semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs

Linghua Chen; Espen R. Jakobsen

doi:10.3934/dcds.2018250

Article Contents

2018, Volume 38, Issue 11: 5735-5763. Doi: 10.3934/dcds.2018250

This issue Previous Article Local correlation entropy Next Article On dispersion decay for 3D Klein-Gordon equation

L¹ semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs

Linghua Chen and
Espen R. Jakobsen^,

Norwegian University of Science and Technology, NO-7491, Trondheim, Norway

* Corresponding author: Espen R. Jakobsen
* Corresponding author: Espen R. Jakobsen

Received: December 31, 2017

Revised: May 31, 2018

Published: August 2018

E. R. Jakobsen is supported by the Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway

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Abstract

We prove a new generation result in $L^1$ for a large class of non-local operators with non-degenerate local terms. This class contains the operators appearing in Fokker-Planck or Kolmogorov forward equations associated with Lévy driven SDEs, i.e. the adjoint operators of the infinitesimal generators of these SDEs. As a byproduct, we also obtain a new elliptic regularity result of independent interest. The main novelty in this paper is that we can consider very general Lévy operators, including state-space depending coefficients with linear growth and general Lévy measures which can be singular and have fat tails.

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Mathematics Subject Classification: 47G20, 47D06, 47D07, 60H10, 35K10, 60G51.

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