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Invariant measures related with Poisson driven stochastic differential equation

Author

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  • Lasota, Andrzej
  • Traple, Janusz

Abstract

A Poisson driven stochastic differential equation generates a semigroup of operators (Pt)t[greater-or-equal, slanted]0 describing the evolution of measures along trajectories and a Markov operator P corresponding to the change of measures from a jump to jump. We show that the semigroup (Pt)t[greater-or-equal, slanted]0 has a finite invariant measure if and only if the operator P has the same property. The main result is applied to problems related with the existence and the dimension of invariant measures.

Suggested Citation

  • Lasota, Andrzej & Traple, Janusz, 2003. "Invariant measures related with Poisson driven stochastic differential equation," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 81-93, July.
    • Handle: RePEc:eee:spapps:v:106:y:2003:i:1:p:81-93
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    Cited by:

    1. Czapla, Dawid & Horbacz, Katarzyna & Wojewódka-Ściążko, Hanna, 2020. "Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2851-2885.
    2. Dong, Zhao & Xu, Tiange & Zhang, Tusheng, 2009. "Invariant measures for stochastic evolution equations of pure jump type," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 410-427, February.
    3. Horbacz, Katarzyna, 2016. "Strong law of large numbers for continuous random dynamical systems," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 70-79.

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