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

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.