Invariant measures for generalized Langevin equations in conuclear space

We investigate existence of an invariant probability measure for the equation dXt=A′Xt+dWt in a conuclear space Φ′, where W is a Wiener process in Φ′ and A generates a semigroup in Φ. 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 Φ=S(Rd) and A=−(−Δ)α/2 (the fractional Laplacian) for 0<α<2. As the corresponding α-stable semigroup does not map S(Rd) into itself, this case needs a separate treatment. We consider two large classes of S′(Rd)-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.