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We consider the stochastic reflection problem associated with a self-adjoint operator A and a cylindrical Wiener process on a convex set K with nonempty interior and regular boundary Σ in a Hilbert space H. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov equation with Neumann boundary condition on Σ. . This reprint differs from the original in pagination and typographic detail. 1 2 V. BARBU, G. DA PRATO AND L. TUBARO T 0(dη(t), X(t) − z(t)) ≥ 0, P-a.s., for all z ∈ C([0, T ]; K). The existence and uniqueness of a solution (X, η) to latter equation was first proven by Cépa in [5]. (See also [3] for a slightly different formulation.) Therefore, under the assumptions of [3] or [5], one can construct a transition semigroup in C(K) by the usual formula P t ϕ(x) = E[ϕ(X(t, x))], t ≥ 0, ϕ ∈ C(K).

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In this paper, we investigate the existence and uniqueness problem for the solutions to a class of semilinear stochastic Volterra equations which arise in the theory of heat conduction with memory effects, where the heat source depends on the solution via a dissipative term. Further, we analyse the asymptotic behaviour of the solution and we prove the existence of a ergodic invariant measure.

The solution Xn to a nonlinear stochastic differential equation of the form dXn(t) + An(t)Xn(t) dt − 1