Banach space valued Ornstein-Uhlenbeck processes indexed by the circle (Q2708585)

Periodic stochastic processes appear in many branches of mathematics and mathematical physics. Their importance stems from the fact that traces of certain semigroups are equal to some integrals over periodic maps. Such representation is used, for example, to provide a probabilistic proof of the Atiyah-Singer theorem. Also, a trace of the Schrödinger group can be represented as a Feynman type integral over a certain Hilbert space of periodic maps. Finally, there is much research towards finding stochastic processes with values in manifold valued periodic maps. NEWLINENEWLINENEWLINEIn this paper, periodic Banach space valued Ornstein-Uhlenbeck processes are considered. Firstly, the authors study the periodic stochastic abstract Cauchy problem NEWLINE\[NEWLINEdX(t)=AX(t)dt+BdW_{H}(t), \;t\in [0,T], \qquad X(0)=X(T),NEWLINE\]NEWLINE which is periodic Itô equation in infinite dimensions. Here \(A\) is the generator of a \(C_{0}\)-semigroup \(S(t)\) on a separable real Banach space, \(W_{H}(t)\) is a suitable cylindrical Wiener process with reproducing kernel Hilbert space \(H,\) and \(B:H\to E\) is a bounded linear operator, where \(E\) is some Banach space. Sufficient conditions for the existence of Gaussian mild solutions are obtained. An explicit formula for covariance of such solutions is derived. Also, sufficient conditions are obtained to guarantee that the mild solution is law-equivalent with the mild solution at time \(T\) of the corresponding stochastic abstract Cauchy problem with zero initial condition.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].