David N. Keck, Mark A. McKibben, "Abstract stochastic integrodifferential delay equations", International Journal of Stochastic Analysis, vol. 2005, Article ID 973980, 31 pages, 2005. https://doi.org/10.1155/JAMSA.2005.275
Abstract stochastic integrodifferential delay equations
We investigate a class of abstract stochastic integrodifferential delay equations dependent upon a family of probability measures in a separable Hilbert space. We establish the existence and uniqueness of a mild solution, along with various continuous dependence estimates and Markov (weak and strong) properties of this solution. This is followed by a convergence result concerning the strong solutions of the Yosida approximations of our equation, from which we deduce the weak convergence of the measures induced by these strong solutions to the measure induced by the mild solution of the primary problem under investigation. Next, we establish the th moment and almost sure exponential stability of the mild solution. Finally, an analysis of two examples, namely a generalized stochastic heat equation and a Sobolev-type evolution equation, is provided to illustrate the applicability of the general theory.
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