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Keywords:
invariant measures; stochastic evolution equations; Hilbert space; compact semigroup; Galerkin approximation; differential equations in Hilbert spaces; $\Omega$-sets
invariant measures; stochastic evolution equations; Hilbert space; compact semigroup; Galerkin approximation; differential equations in Hilbert spaces; $\Omega$-sets
Summary:
A differential equation is a Hilbert space with all solutions bounded but with so finite nontrivial invariant measure is constructed. In fact, it is shown that all solutions to this equation converge weakly to the origin, nonetheless, there is no stationary point. Moreover, so solution has a non-empty $\Omega$-set.
References:
[1] G. Da Prato D. Gątarek J. Zabczyk: Invariant measures for semilinear stochastic equations. Stochastic Anal. Appl. 10 (1992), 387-408. DOI 10.1080/07362999208809278 | MR 1178482
[2] N. N. Vakhaniya V. I. Tarieladze S. A. Chobanyan: Probability distributions in Banach spaces. Nauka, Moscow, 1985. (In Russian.) MR 0787803
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