Invariant measures for stochastic evolution equations with Hilbert space valued Lévy noise

Existence of invariant measures for semi-linear stochastic evolution equations in separable real Hilbert spaces is considered, where the noise is generated by Hilbert space valued Lévy processes. It is shown that if the Lévy process has locally bounded second moments, if the semigroup generated by the linear part is hyperbolic, and if the Lipschitz constants of the nonlinearities are sufficiently small, then existence of a mean square bounded solution implies existence of an invariant measure. In case the semigroup is exponentially stable, each solution is mean square bounded and there exists a unique invariant measure with finite second moment whenever the Lipschitz constants of the nonlinearities are sufficiently small. The stochastic integral with respect to the Hilbert space valued Lévy process is constructed as a series by means of a decomposition of the process into scalar processes. The existence of the invariant measure is proved by a coupling argument, which depends on weak uniqueness of solutions of the equation