For many years the attention of researchers has been attracted by mathematical models of various objects of rather complicated nature, the evolution of which occurs in the field of random forces, taking into account the aftereffect. The most common of such models are described by stochastic functional differential evolution partial differential equations. In contrast to the classical stochastic differential equations, which can be named «ordinary», these equations combine the features of functional differential equations in partial derivatives and Ito stochastic equations. Interest in such equations has arised almost simultaneously in the theory of partial differential equations and in the theory of random processes. Number of works is devoted to the study of solutions of such equations of different stochastic nature in finite-dimensional and various infinite-dimensional functional spaces. The theory of stochastic partial differential equations is an important direction in the development of the modern theory of stochastic equations.
In the thesis initial-value problems for stochastic partial functional-differential equations of reaction-diffusion type, both with variable and with constant function of delay, are studied. Such equations describe the evolution in time and in the whole space of processes with values in Hilbert spaces. The solution is understood in the «mild» sense of the mild solution, that is, in terms of special Hilbert-valued operators that generate it.
The thesis consists of the annotations in Ukrainian, English and Russian, the list of symbols, introduction, four sections of the main part, conclusions, references and two appendices.
The first section contains a review of literature on the subject of the thesis work.
In the second section of the thesis, initial-value problems for stochastic integro-differential reaction-diffusion equations of neutral type with variable delay are studied. The purpose of the second section is to investigate the correct solvability of initial-value problems for equations of such type: the existence, uniqueness and continuity from initial datum of their mild solutions, - in terms of the coefficients of the equation and some properties of their solutions.
The third section is devoted to obtaining an analogue of the theorem from the pioneering work of A. V. Skorokhod – a comparison theorem for the solutions of the initial-value problem for two Ito stochastic differential equations with the same diffusion coefficients. In this section the initial-value problem for two stochastic integro-differential reaction-diffusion equations of neutral type with a constant delay. The purpose of the third section is to prove the comparison theorem for two mild solutions of this initial-value problem via using ideas from the work of R. Manthey and T. Zausinger and pioneering work of A. V. Skorokhod.
In the fourth section of the thesis the asymptotic behavior of solutions to the initial-value problem for the stochastic differential equation with delay at infinity is studied in sense of existence for it invariant measures in a special Hilbert space. The purpose of the fourth section is to realize a well-known compactness approach for finding the coefficient conditions for the existence of probability invariant measure. The basis of this approach is the well-known Krylov-Bogolyubov theorem on the existence of an invariant measure for an abstract Markov dynamical system. The main ideas from this approach are applied to the problem under investigation. In order to implement this approach, it is necessary to prove the existence, uniqueness and continuity from initial datum of the solution, Markov property, Feller property, the compactness of the semigroup, corresponding to the equation, and, most importantly, the existence of globally bounded in probability solution in a suitably selected Hilbert space.
The first appendix contains the proof of some auxiliary assertions. The second appendix contains the list of publications on the topic of the thesis (twenty titles) and information about the approbation of the results of the thesis.