Ergodic properties of recurrent solutions of stochastic evolution equations (Q1804699)

Ergodic and recurrence properties for a semilinear stochastic equation of the general form \[ dX(t) = (AX(t) + f(X(t)))dt + dW_ t, \tag{1} \] in a real separable Hilbert space \(H\) are investigated, where the linear operator \(A\) generates a strongly continuous semigroup on \(H\), \(W_ t\) is a cylindrical Wiener process on \(H\) with a bounded covariance operator and \(f\) is a nonlinear function defined on a subspace of \(H\). The aim of the paper is to study \(\sigma\)-finite and probabilistic invariant measures for the equation (1). Under suitable conditions (roughly speaking, sufficient nondegeneracy of the noise and a recurrence condition) the ratio ergodic theorem and some of its consequences as well as the existence and uniqueness of a \(\sigma\)-finite invariant measure are shown. In such cases, there is a dichotomy in behavior of the solutions to (1), analogous to the one in the finite-dimensional case.