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 Electronic Communications in Probability > Vol. 11 (2006) > Paper 3 open journal systems 


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.


Full text: PDF

Pages: 24-34

Published on: March 29, 2006
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Electronic Communications in Probability. ISSN: 1083-589X