Invariant measures for generalized Langevin equations in conuclear space

We investigate existence of an invariant probability measure for the equation in a conuclear space [Phi]', where W is a Wiener process in [Phi]' and generates a semigroup in [Phi]. In the first part of the paper we formulate a sufficient and necessary condition for the existence of an invariant measure and we describe all invariant measures. In the second part we investigate the case and (the fractional Laplacian) for 0<[alpha]<2. As the corresponding [alpha]-stable semigroup does not map into itself, this case needs a separate treatment. We consider two large classes of -Wiener processes: those determined by homogeneous random fields and those associated with tempered kernels. In both cases, we formulate conditions which are sufficient (and, in a sense, necessary or almost necessary) for the existence of stationary measures, and we give several examples.