Invariant measures for generalized Langevin equations in conuclear space

We investigate existence of an invariant probability measure for the equati on dX(t) = A' X-t + dW(t) in a conuclear space Phi', where W is a Wiener pr ocess in Phi' and A generates a semigroup in Phi. In the first part of the paper we formulate a sufficient and necessary condition for the existence o f an invariant measure and we describe all invariant measures. In the secon d part we investigate the case Phi = S(R-d) and A = -(-Delta)(alpha/2) (the fractional Laplacian) for 0 < alpha < 2. As the corresponding alpha-stable semigroup does not map S([R-d) into itself, this case needs a separate tre atment. We consider two large classes of S'(R-d)-Wiener processes: those de termined by homogeneous random fields and those associated with tempered ke rnels. In both cases, we formulate conditions which are sufficient (and, in a sense, necessary or almost necessary) for the existence of stationary me asures, and we give several examples. (C) 1999 Elsevier Science B.V. All ri ghts reserved.