The thesis is devoted to the study of infinite-dimensional stochastic functional- differential equations in Hilbert spaces, which are mathematical models. the most diverse objects of a complex nature, the evolution of which takes place in the field of random forces, taking into account the after effect. The most common among such models are described by stochastic partial functional-differential equations.Unlike classical stochastic differential equations, which can be called "ordinary these equations combine the features of functional differential equations with partial derivatives and stochastic Ito equations. Interest in these equations arose almost simultaneously in the theory of equations with partial derivatives and in the theory of random processes.
A large number of works are devoted to the study of solutions of such equations of various stochastic nature finite-dimensional and various infinite-dimensional functional spaces. Since the majority of modern mathematical models describe processes with distributed parameters, stochastic equations with partial derivatives, or more broadly, equations with unbounded operators, acquire special importance. The theory of stochastic differential equations with unbounded operators is an important direction in the development of the modern theory of stochastic equations.
We stude the initial value problems for stochastic functional-differential equations of both ordinary and neutral types, that is, when the delay effect is manifested not only in the coefficients of the equation, but also in the "derivative". For such equations, the conditions for the existence and uniqueness of the solution were obtained, their continuous dependence on the initial data was studied, and the Markov and Feller properties of the solutions in displacement spaces were established. At the same time, different approaches to defining the solution are considered: mild, weak and strong.
When proving the existence of a soft solution, the apparatus of the analytical theory of semigroups of bounded operators generated by the unbounded operator included in the right-hand side of the equation is used. At the same time, the properties
of stochastic convolution, that is, stochastic convolution of the corresponding semigroup with the coefficients of the right-hand side of the equation, are significantly used. This approach was widely used in the study of infinite-dimensional stochastic systems without delay in the works of G. Da Prato, J. Zabczyk, S. Cerrai, M. Hairer and other authors. For stochastic functional differential equations, it is also widely used in the works of T. Govindan, Q. Li, M. Wei and other authors. However, for equations of neutral types, similar results are obtained only under fairly strict assumptions. The latter is caused by the presence of an unbounded operator in the soft solution formula. Another important aspect is that real mathematical models are equations in which the right-hand sides are interpreted as external influences, which do not have to be smooth, even Lipschitz functions. Therefore, the question arises of establishing the conditions for the existence and unity of solutions without the Lipshitz condition and linear growth. This is exactly the case that is studied in the work.
Establishing the conditions for the existence of weak solutions is carried out using the theory of monotone operators, as well as using the compactness approach developed at the Lyons school. The adaptation of these approaches to stochastic equations is carried out in the works of Huang L, Mao X, Wei Liu, Michael Rockner and other authors. However, for functional differential equations in this direction, results are obtained only in some partial cases. It is important to note that the Lipshitz condition is not imposed on the right-hand side, which is replaced by a certain condition of monotonicity and exponential growth.
The existence of strong solutions was previously considered only for equations with a fixed delay.
A dissertation study is devoted to filling these gaps. In particular, theorems for the existence of soft solutions for equations of the neutral type were obtained under much weaker conditions than those of the above-mentioned authors, and the existence of weak solutions for coupled equations was proved, one of which is an infinite¬dimensional stochastic functional-differential, and the other is an ordinary differential. Such equations appear in various applications: for example, two-domain equations (defibrillator model), Hodgkin-Huxley equation for a nerve axon, nuclear dynamics equation, and others. When establishing the conditions for the existence of strong solutions, an approach based on obtaining a priori estimates of the mathematical expectation of various Sobolev-type norms with subsequent application of Sirrin-type theorems was used.
