Invariant Measures for Stochastic Cauchy Problems with Asymptotically Unstable Drift Semigroup

We investigate existence and permanence properties of invariant measures for abstract stochastic Cauchy problems of the form dU(t) = (AU(t) + f) dt+B dWH(t), t > 0, governed by the generator A of an asymptotically unstable C0-semigroup on a Banach space E. Here f ∈ E is fixed, WH is a cylindrical Brownian motion over a separable real Hilbert space H, and B : H → E is a bounded operator. We show that if c0 6⊆ E, 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. RESEARCH SUPPORTED BY THE RESEARCH TRAINING NETWORK HPRN-CT-2002-00281 RESEARCH SUPPORTED BY THE NWO VIDI SUBSIDIE 639.032.201 AND THE RESEARCH TRAINING NETWORK HPRN-CT-2002-00281 24 Invariant measures 25