The article presents a detailed scheme of mean-square approximation for evolutionary stochastic equations with delay in infinite-dimensional spaces. The main attention is paid to replacing the initial system with an aftereffect by an evolutionary system of stochastic equations without aftereffect. The proposed approach involves dividing the delay interval into subintervals and constructing the corresponding system of equations that approximates the behavior of the initial system. It is important to note that the number of equations in such an approximating system increases with the number of subintervals. The main result of the study shows that under the condition that the partition becomes smaller and smaller (i.e. the number of subintervals tends to infinity), the mean square distance between the solutions of the equation with delay and the solutions of the system without delay tends to zero.

The theoretical basis of the approximation method uses key concepts and results from stochastic analysis in infinite-dimensional spaces, in particular, to solve problems related to the functional nature of the aftereffect and the unboundedness of the state space. The study not only generalizes previous results for finite-dimensional cases to the infinite-dimensional environment, but also extends the methods used for deterministic equations with delay to stochastic systems. The methodology is based on the classical idea of ​​the expansion of the solution of the equation with delay according to the Taylor formula in the length of the delay interval h > 0 h>0. This approach allows replacing the initial problem for the delay equation with a system of Cauchy problems for a system of ordinary differential equations constructed in a special way.

The results of the work have significant practical implications, especially for systems where delays are natural, such as stochastic control systems, population dynamics, or infinite-dimensional systems described by stochastic partial differential equations. By replacing complex systems with delays with simpler systems without delays, the proposed method not only simplifies numerical calculations, but also provides a deeper understanding of the dynamics of such systems. Proving the conditions under which the approximation is correct contributes to the development of the theoretical basis of stochastic equations with delays in infinite-dimensional spaces and offers a powerful tool for their analysis and modeling.

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