Stability in Distribution of Neutral Stochastic Functional Differential Equations with Infinite Delay

TL;DR

This paper studies the stability in distribution of neutral stochastic functional differential equations with infinite delay, establishing conditions for existence, uniqueness, and stability of solutions in a specific weighted function space.

Contribution

It introduces a strong monotone condition ensuring existence, uniqueness, and stability of solutions for NSFDEwID in an infinite delay setting.

Findings

Established a sufficient condition for solution existence and uniqueness.

Proved stability in distribution of solutions under the given conditions.

Provided an illustrative example demonstrating the theory.

Abstract

In this paper, we investigate stability in distribution of neutral stochastic functional differential equations with infinite delay (NSFDEwID) at the state space \begin{equation*} C_{r}=\{{\varphi\in C((-\infty,0];R^{d}):\|\varphi\|_{r}=\sup_{-\infty<\theta\leq0}e^{r\theta}\lvert\varphi(\theta)\rvert} < \infty\ , \quad r > 0 \}. \end{equation*} We drive a sufficient strong monotone condition for the existence and uniqueness of the global solutions of NSFDEwID in the state space $ C_{r} $. We also address the stability of the solution map $ x_{t} $ and illustrate the theory with an example.