Banach space valued Ornstein-Uhlenbeck processes indexed by the circle

In this paper we study the periodic stochastic abstract Cauchy problem dX(t) = AX(t)dt + BdW(H)(T), t is an element of [0,T], X(0) = X(T), where A is the generator of a C-0-semigroup {S(t)}(t greater than or equal to0) on a separable real Banach space, {W-H(t)}(t greater than or equal to0) is a suitable cylindrical Wiener process with reproducing kernel Hilbert space H, and B : H --> E is a bounded linear operator. We obtain sufficient conditions for existence of Gaussian mild solutions and show that solutions and compute the covariance of these solutions. We also obtain sufficient conditions which guaratee 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.