On Comparison Results for Neutral Stochastic Differential Equations of Reaction-Diffusion Type in L2(ℝd)

Abstract

In the present paper, we establish a comparison result for solutions to the Cauchy problems for two stochastic integro-differential equations of reaction-diffusion type with delay. On this subject number of authors have obtained their comparison results. We deal with the Cauchy problems for two stochastic integro-differential equations of reaction-diffusion type with delay. Except drift and diffusion coefficients, our equations include also one integro-differential term. Basic difference of our case from the case of all earlier investigated problems is presence of this term. Presence of this term turns this equation into a nonlocal neutral stochastic equation of reaction-diffusion type. Nonlocal deterministic equations of this type are well known in literature and have wide range of applications. Such equations arise, for instance, in mechanics, electromagnetic theory, heat flow, nuclear reactor dynamics, and population dynamics. These equations are used in modeling of phytoplankton growth, distant interactions in epidemic models and nonlocal consumption of resources. We introduce a concept of mild solutions to our problems and state and prove a comparison theorem for them. According to our result, under certain assumptions on coefficients of equations under consideration, their solutions depend on the transient coefficients in a monotone way.