Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps.

*(English)*Zbl 1186.60057Summary: The existence, uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented. The principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stability of strong solutions to mild ones. As a consequence, stability results of G. K. Basak, A. Bisi and M. K. Ghosh [J. Math. Anal. Appl. 202, 604-622 (1996; Zbl 0856.93102)] and Ch. Yuan et al. [Syst. Control Lett. 50, No. 3, 195–207 (2003; Zbl 1157.60330)] are generalized to cover a class of much more general stochastic partial differential delay equations with jumps in infinite dimensions. In contrast to the almost sure exponential stability by A. Ichikawa [J. Math. Anal. Appl. 90, 12–44 (1982; Zbl 0497.93055)] and J. Luo and K. Liu [Stochastic Processes Appl. 118, No. 5, 864–895 (2008; Zbl 1186.93070)] and the moment exponential stability in Luo & Liu, we present a new result on the stability in distribution of mild solutions. Finally, an example is given to demonstrate the applicability of our work.

##### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

34K30 | Functional-differential equations in abstract spaces |

##### Keywords:

stochastic partial differential delay equation; mild solution; stability in distribution; jump processes##### References:

[1] | 202 pp 604– (1999) |

[2] | STOCH ANAL APPL 17 pp 743– (1999) |

[3] | PROC R SOC LOND A 456 pp 1775– (2000) |

[4] | J THEOR PROBAB 21 pp 322– (2008) |

[5] | STOCH ANAL APPL 21 pp 1059– (2003) |

[6] | 90 pp 12– (1982) |

[7] | J MATH KYOTO UNIV 181 pp 749– (2002) |

[8] | STOCH PROC APPL 118 pp 864– (2008) |

[9] | POTENTIAL ANAL 26 pp 255– (2007) |

[10] | 331 pp 191– (2007) |

[11] | 181 pp 72– (2002) |

[12] | STATIST PROBAB LETT 78 pp 490– (2007) |

[13] | STOCH PROC APPL 103 pp 277– (2003) |

[14] | SYST CONTROL LETT 50 pp 195– (2003) |

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