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Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps. (English) Zbl 1186.60057
Summary: 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 {\it G. K. Basak, A. Bisi} and {\it M. K. Ghosh} [J. Math. Anal. Appl. 202, 604-622 (1996; Zbl 0856.93102)] and {\it 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 {\it A. Ichikawa} [J. Math. Anal. Appl. 90, 12--44 (1982; Zbl 0497.93055)] and {\it J. Luo} and {\it 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.

60H15Stochastic partial differential equations
34K30Functional-differential equations in abstract spaces
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