Atomic operators, random dynamical systems and invariant measures

Ponosov, A.; Stepanov, E.

doi:10.1090/spmj/1353

Atomic operators, random dynamical systems and invariant measures
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by A. Ponosov and E. Stepanov

St. Petersburg Math. J. 26 (2015), 607-642

DOI: https://doi.org/10.1090/spmj/1353

Published electronically: May 6, 2015

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Abstract:

It is proved that the existence of invariant measures for families of the so-called atomic operators (nonlinear generalized weighted shifts) defined over spaces of measurable functions follows from the existence of appropriate invariant bounded sets. Typically, such operators come from infinite-dimensional stochastic differential equations generating not necessarily regular solution flows, for instance, from stochastic differential equations with time delay in the diffusion term (regular solution flows called also Carathéodory flows are those almost surely continuous with respect to the initial data). Thus, it is proved that to ensure the existence of an invariant measure for a stochastic solution flow it suffices to find a bounded invariant subset, and no regularity requirement for the flow is necessary. This result is based on the possibility to extend atomic operators by continuity to a suitable set of Young measures, which is proved in the paper. A motivating example giving a new result on the existence of an invariant measure for a possibly nonregular solution flow of some model stochastic differential equation is also provided.

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