Invariant measures of fractional stochastic delay reaction-diffusion equations on unbounded domains. (English) Zbl 1473.37097
Summary: In this paper, existence of invariant measure is mainly investigated for a fractional stochastic delay reaction-diffusion equation defined on unbounded domains. We first establish the mean-square uniform smallness of the tails of the solutions in order to overcome the non-compactness of standard Sobolev embeddings on unbounded domains. We then show the weak compactness of a family of probability distributions of the solutions by combining the Ascoli-Arzelà theorem, the uniform tail-estimates as well as the technique of dyadic division.
MSC:
| 37L55 | Infinite-dimensional random dynamical systems; stochastic equations |
| 37L40 | Invariant measures for infinite-dimensional dissipative dynamical systems |
| 35K57 | Reaction-diffusion equations |
| 35R11 | Fractional partial differential equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
Keywords:
invariant measure; fractional reaction-diffusion equation; unbounded domain; stochastic delay equationReferences:
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