Exponential ergodicity of semilinear equations driven by Lévy processes in Hilbert spaces

Anna Chojnowska-Michalik and Ben Goldys

Abstract

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 [12]. In the proof we use an idea of Maslowski and Seidler from [10].

Keywords: Lévy noise, semilinear SPDE, exponential ergodicity, total variation.

AMS Subject Classification: Primary 60G51; secondary 60H15.