Communications in Mathematical Sciences

Volume 15 (2017)

Number 4

Invariant measures for SDEs driven by Lévy noise: A case study for dissipative nonlinear drift in infinite dimension

Pages: 957 – 983

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n4.a3

Authors

Sergio Albeverio (Department of Applied Mathematics, University of Bonn, Germany)

Luca di Persio (Department of Computer Science, University of Verona, Italy)

Elisa Mastrogiacomo (Department of Statistics and Quantitive Methods, University of Milano Bicocca, Milano, Italy)

Boubaker Smii (Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia)

Abstract

We study a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by Lévy noise. We define a Hilbert–Banach setting in which we prove existence and uniqueness of solutions under general assumptions on the drift and the Lévy noise. We then prove a decomposition of the solution process into a stationary component, the law of which is identified with the unique invariant probability measure μ of the process, and a component which vanishes asymptotically for large times in the $L^p (/mu)$-sense, for all $1 \leq p \lt {+ \infty}$.

Keywords

nonlinear SPDEs, dissipative nonlinear drift, Lévy noise, invariant measure

2010 Mathematics Subject Classification

35S05, 37L40, 47H06, 60G10, 60J75

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Published 16 May 2017