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Exponential ergodicity of semilinear equations driven by Lévy processes in Hilbert spaces. (English) Zbl 1323.60082

Chojnowska-Michalik, Anna (ed.) et al., Stochastic analysis. Special volume in honour of Jerzy Zabczyk. Selected papers based on the presentations at the Banach Center conference on stochastic analysis and control, Bȩdlewo, Poland, May 6–10, 2013. Warsaw: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-28-7/pbk). Banach Center Publications 105, 59-72 (2015).
Summary: 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 E. Priola and J. Zabczyk [Probab. Theory Relat. Fields 149, No. 1–2, 97–137 (2011; Zbl 1231.60061)] we generalize results from [E. Priola et al., Stochastic Processes Appl. 122, No. 1, 106–133 (2012; Zbl 1239.60059)]. In the proof we use an idea of B. Maslowski and J. Seidler [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 10, No. 2, 69–78 (1999; Zbl 1007.60067)].
For the entire collection see [Zbl 1323.60004].

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
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