Uniqueness of invariant measures for the stochastic Cauchy problem in Banach spaces. (English) Zbl 0986.60055
Kérchy, László (ed.) et al., Recent advances in operator theory and related topics. The Béla Szőkefalvi-Nagy memorial volume. Proceedings of the memorial conference, Szeged, Hungary, August 2-6, 1999. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 127, 491-517 (2001).
Let \(A\) be a generator of a \(C_0\)-semigroup \((S_{t})\) on a real separable Banach space \(E\), \(W\) a standard cylindrical Wiener process on a real separable Hilbert space \(H\) and \(B\in{\mathcal L}(H,B)\) a bounded linear operator. Define an operator \(Q_{t}\in{\mathcal L} (E^*,E)\) by \(Q_{t}x^* = \int^{t}_{0} S_{s}BB^*S^*_{s} x^* ds\), \(x^*\in E^*\). Z. Brzeźniak and the author [Stud. Math. 143, No. 1, 43-74 (2000; Zbl 0964.60043)] showed that if \(Q_{t}\) is a covariance operator of a centered Gaussian Borel measure on \(E\) for every \(t>0\), then for any \(x\in E\) there exists a unique weak solution to a linear stochastic differential equation
\[
du = Au dt + B dW,\qquad u(0) = x,\tag{1}
\]
and (1) defines a Markov process on \(E\). Suppose moreover that there exists an operator \(Q_\infty\in{\mathcal L}(E^*,E)\) such that \(\langle Q_\infty x^*,y^*\rangle = \lim_{t\to\infty}\langle Q_{t} x^*,y^*\rangle\) for all \(x^*,y^*\in E^*\) and \(Q_\infty\) is a covariance operator of a centered Gaussian Borel measure \(\mu _\infty\) on \(E\). Then \(\mu_\infty\) is an invariant measure for this Markov process. Several interesting criteria for the measure \(\mu_\infty\) to be the unique invariant Gaussian Borel measure on \(E\) are given. For example, uniqueness holds provided either \((S_{t})\) is strongly stable, or the support of \(\mu_\infty\) equals to the whole space \(E\) and \((S_{t})\) is compact, or if the support of \(\mu_\infty\) is \(E\) and \(t\mapsto S^*_{t}x^*\) is bounded for all entire vectors \(x^*\in \bigcap_{n\geq 1}\text{Dom}(A^{*n})\). The proofs are purely functional-analytic and they provide a new insight even in the classical case of a Hilbert space \(E\).
For the entire collection see [Zbl 0971.00017].
For the entire collection see [Zbl 0971.00017].
Reviewer: Jan Seidler (Praha)
MSC:
60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
60J35 | Transition functions, generators and resolvents |
93C25 | Control/observation systems in abstract spaces |
47D03 | Groups and semigroups of linear operators |
35R15 | PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) |