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].