Ultimate Boundedness and Invariant Measure

Abstract

We study ultimate boundedness in the m.s.s. of solutions to SDE’s, namely the following property, $E\| X^{x}(t)\|^{2}_{H}\leq c\mathrm{e}^{-\beta t}\| x\|_{H}^{2}+M$, $c,\; \beta >0$, M<∞.

The methods used here are similar to those employed in the study of exponential stability in the m.s.s. and involve obtaining necessary and sufficient conditions for ultimate boundedness in the m.s.s. in terms of the Lyapunov function. We achieve this goal both for mild and for strong variational solutions. As a consequence we obtain sufficient conditions for ultimate boundedness in the m.s.s. in terms of exponential stability of a solution of the related deterministic problem. We also obtain conditions for non-linear equations in terms of the behavior of their coefficients at infinity (in norm). Our results are applied to study the behavior of solutions for specific examples of SPDE’s.

Next, we show that under the assumption of ultimate boundedness in the m.s.s. there exists an invariant measure for mild solutions and for strong variational solutions in a Gelfand triplet V↪H↪V ^∗, provided the embeddings are compact. Under this compactness assumption we prove the recurrance of strong variational solutions to a compact set. Finally, we examine asymptotic behavior of solutions to SPDE’s such as the heat equation, Navier–Stokes equation, and of mild solutions to semilinear equations driven by a cylindrical Wiener process.