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Invariant measures for stochastic Cauchy problems with asymptotically unstable drift semigroup. (English) Zbl 1111.60044

Summary: We investigate existence and permanence properties of invariant measures for abstract stochastic Cauchy problems of the form \(dU(t) = (AU(t) + f) dt + B dW_H(t)\), 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.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34G10 Linear differential equations in abstract spaces
35R60 PDEs with randomness, stochastic partial differential equations
47D07 Markov semigroups and applications to diffusion processes
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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