Explicit invariant measures for infinite dimensional SDE driven by Lévy noise with dissipative nonlinear drift I

Abstract

We study a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by Lévy noise. Our work is divided in two parts. In the present part I we first define a Hilbert-Banach setting in which we can 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 in a stationary component and a component which vanishes asymptotically for large times in the L p −sense, p ≥ 1. The law of the stationary component is identified with the unique invariant probability measure of the process.In part II we will exhibit the invariant measure as the limit of explicit invariant measures for finite dimensional approximants.We shall present a general discussion of such explicit invariant measures involving, in particular, ground state transformation for Lévy driven processes and relations to invariant measures for Ornstein-Uhlenbeck like processes with Lévy noise.Examples and applications are also provided.

A Class of Lévy Driven SDEs and their Explicit Invariant Measures

“…This paper is devoted to the search of such explicit measures for (in)finite dimensional SDE driven by Lévy noise and with nonlinear drift coefficients. This connects to our previous paper [8], where we studied such equations in the infinite dimensional case. In that paper we found, in particular, abstract invariant probability measures for the equations at hand and we discussed their relations with a decomposition of the solution process as a sum of a stationary component and an asymptotically in time vanishing component.…”

“…In chapter 3 we discuss the infinite dimensional case. Section 3.1 presents the case of an infinite dimensional O-U Lévy-process, following basic work by [55], stressing also the relation with our paper [8]. Section 3.2 presents the case of certain infinite dimensional Lévy driven systems , which can be seen as infinite dimensional limits of the finite dimensional systems discussed in Section 2.4.…”

Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes

“…the Ornstein-Uhlenbeck process), may be an important first step in obtaining information about the solution to the more general equation, see e.g. [45,2]. The two different approaches to solving SPDEs that we have presented above, using on the one hand, two-parameter "space-time white noise", and on the other hand, infinite-dimensional processes, have each generated a considerable literature; nonetheless there are important cases where they both give rise to the same solution, see e.g.…”

A Graphical Representation of the Truncated Moment of the Solution of a Nonlinear SPDE

“…The dynamics of many phenomena studied in sciences, engineering and economy are described in many cases by stochastic differential equations (SDEs), see for example [1,2,6]. Stochastic partial differential equations (SPDE) is an interesting class of SDEs with numerous applications in different fields, which received a lot of attention during the last 50 years.…”