Invariant measures for stochastic Cauchy problems with asymptotically unstable drift semigroup
Onno van Gaans, Leiden University Jan van Neerven, Technical University of Delft
Abstract
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),
governed by the generator A of
an asymptotically unstable C0-semigroup on a Banach space E.
Here f in E is fixed, WH 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 c0, 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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