Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions |
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Authors: | Kai Liu |
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Institution: | 1. Division of Statistics and Probability, Department of Mathematical Sciences , The University of Liverpool , Liverpool, United Kingdom k.liu@liv.ac.uk |
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Abstract: | Abstract In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established. |
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Keywords: | Fundamental solutions Stochastic convolutions Stochastic retarded differential equations Strong weak and mild solutions |
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