Journal of Applied Mathematics and Stochastic Analysis
Volume 2005 (2005), Issue 3, Pages 275-305
doi:10.1155/JAMSA.2005.275
Abstract stochastic integrodifferential delay equations
1Department of Mathematics, Ohio University, Athens 45701, OH, USA
2Mathematics and Computer Science Department, Goucher College, Baltimore 21204, MD, USA
Received 26 August 2004; Revised 2 November 2004
Copyright © 2005 David N. Keck and Mark A. McKibben. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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 pth 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.