Existence and Uniqueness Results, the Markovian Property of Solution for a Neutral Delay Stochastic Reaction-Diffusion Equation in Entire Space

Abstract

We establish results, concerning existence, uniqueness, and continuous dependence on an initial datum of a mild solution for neutral stochastic integro-differential equations with variable time delay of reaction-diffusion type. We also establish the Markovian property of this solution. Herewith our emphasis is on unbounded domain {x ∈ R d }.

Invariant measure for neutral stochastic functional differential equations with non-Lipschitz coefficients

“…Furthermore, following [9] introduce the interpolation space D A ( 1 2 , 2) = H 1 0 . Therefore by Proposition A.17 [14]…”

“…In order to apply the compactness approach, we need to establish that the solutions are both Markovian and Feller. These properties were established for (1) in [14] in case when A is Laplacian. The analog of comparison theorem for such equations was established in [30].…”

Invariant Measure for Neutral Stochastic Functional Differential Equations with Non-Lipschitz Coefficients

“…In order to apply the compactness approach, we need to establish that the solutions are both Markovian and Feller. These properties were established for (1) in [14] in case when A is Laplacian. The analog of comparison theorem for such equations was established in [30].…”

On Weak and Strong Solutions of Paired Stochastic Functional Differential Equations in Infinite-Dimensional Spaces

“…The work [4] is devoted to the existence of weak solutions. Similar equations of neutral type have been studied in [5], [6], [7], [8], [9], where the questions of controllability and the Markovian property have been considered and comparison theorems have been obtained. In [10], using the compactness approach [11] the authors establish the existence of an invariant measure.…”