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2006 Invariant measures for stochastic Cauchy problems with asymptotically unstable drift semigroup
Onno Gaans, Jan Neerven
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Electron. Commun. Probab. 11: 24-34 (2006). DOI: 10.1214/ECP.v11-1184


We investigate existence and permanence properties of invariant measures for abstract stochastic Cauchy problems of the form $$dU(t) = (AU(t)+f)\,dt + B\,dW_H(t), \ \ t\ge 0,$$ governed by the generator $A$ of an asymptotically unstable $C_0$-semigroup on a Banach space $E$. Here $f \in E$ is fixed, $W_H$ is a cylindrical Brownian motion over a separable real Hilbert space $H$, and $B$ is a bounded operator from $H$ to $E$. We show that if $E$ does not contain a copy of $c_0$, such invariant measures fail to exist generically but may exist for a dense set of operators $B$. It turns out that many results on invariant measures which hold under the assumption of uniform exponential stability of $S$ break down without this assumption.


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Onno Gaans. Jan Neerven. "Invariant measures for stochastic Cauchy problems with asymptotically unstable drift semigroup." Electron. Commun. Probab. 11 24 - 34, 2006.


Accepted: 29 March 2006; Published: 2006
First available in Project Euclid: 4 June 2016

zbMATH: 1111.60044
MathSciNet: MR2211424
Digital Object Identifier: 10.1214/ECP.v11-1184

Primary: 35R15
Secondary: 47D06 , 60H05

Keywords: Invariant measures , stochastic evolution equations in Hilbert spaces


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