Exponential ergodicity of semilinear equations driven by Lévy processes in Hilbert spaces
We study convergence to the invariant measure for a class of semilinear stochastic evolution equations driven by Lévy noise, including the case of cylindrical noise. For a certain class of equations we prove the exponential rate of convergence in the norm of total variation. Our general result is applied to a number of specific equations driven by cylindrical symmetric $\alpha$-stable noise and/or cylindrical Wiener noise. We also consider the case of a “singular” Wiener process with unbounded covariance operator. In particular, in the equation with diagonal pure $\alpha$-stable cylindrical noise introduced by Priola and Zabczyk we generalize results from Priola, Shirikyan, Xu and Zabczyk (2012). In the proof we use an idea of Maslowski and Seidler (1999).