Systems of stochastic functional differential equations (Q2784243)

The thesis is devoted to the investigation of stochastic partial functional differential equations and stochastic evolution equations. The problem of existence and uniqueness as well as qualitative properties are studied. After an introduction to preliminaries of the corresponding deterministic theory and to infinite-dimensional stochastic analysis, especially the theory of stochastic integration, a class of stochastic evolution equations in \(L_2\) is considered. Here the coefficients are supposed to be Lipschitz continuous and bounded. The linear operator in the drift may be time-dependent and generates a strongly continuous two-parameter semigroup. Existence, uniqueness, continuity, nonnegativity and boundedness of corresponding mild and strong solutions are shown. Moreover, a comparison result is proved.NEWLINENEWLINENEWLINESystems of parabolic stochastic partial functional differential equations which cover generalizations of models of population dynamics like predator-prey models and competition models with hereditary terms are considered in the next part. The solutions are pathwise continuous random fields defined in the mild sense. Sufficient conditions for existence, uniqueness, nonnegativity and boundedness of the solutions are given. Moreover, stability and comparison results are proved. Finally, relations between the stochastic evolution equations in \(L_2\) considered above and stochastic partial differential equations are investigated. The thesis also contains a result on the existence of pathwise continuous modifications for vector valued random fields.