Invariant measure for the stochastic Cauchy problem driven by a cylindrical Lévy process

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Abstract

In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process, and show that these conditions are also necessary if the semigroup is stable, in which case the invariant measure is unique. For typical situations such as the heat equation, we significantly simplify these conditions without assuming any further restrictions on the driving cylindrical Lévy process and demonstrate their application in some examples.
Original languageEnglish
Number of pages31
JournalJournal of Mathematical Analysis and Applications
Publication statusAccepted/In press - 20 Aug 2020

Keywords

  • cylindrical Lévy processes, Cauchy problem, invariant measures, stationary distributions, Mehler semigroup

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