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An ergodic theorem of a parabolic Anderson model driven by Lévy noise. (English) Zbl 1272.60046

We study the ergodic theorem of a parabolic Anderson model driven by Levy noise. Under the assumption that \(A\) is symmetric with respect to a \(\sigma\)-finite measure \(\pi\), we obtain the long-time convergence to an invariant probability measure starting from a bounded nonnegative \(A\)-harmonic function based on the self-duality property. Furthermore, under some mild conditions, we obtain the one-to-one correspondence between the bounded nonnegative \(A\)-harmonic functions and the extremal invariant probability measures with finite second moment of the nonnegative solution of the parabolic Anderson model driven by Levy noise.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60B10 Convergence of probability measures
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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