New well-posedness results for stochastic delay Rayleigh-Stokes equations

In this work, the following stochastic Rayleigh-Stokes equations are considered\begin{document}$ \begin{align*} \partial_t \big[ x(t)+f(t,x_\rho(t)) \big] = \big( A +\vartheta &\partial_t^\beta A \big) \big[ x(t)+f(t,x_\rho(t)) \big] \\ &+ g(t,x_\tau(t)) + B(t,x_\xi(t)) \dot{W}(t), \end{align*} $\end{document}which involve the Riemann-Liouville fractional derivative in time, delays and standard Brownian motion. Under two different conditions for the non-linear external forcing terms, two existence and uniqueness results for the mild solution are established respectively, in the continuous space \begin{document}$ \mathcal{C}([-h,T];L^p(\Omega,V_q)) $\end{document} , \begin{document}$ p \ge 2 $\end{document} , \begin{document}$ q \ge 0 $\end{document} . Our study was motivated and inspired by a series of papers by T. Caraballo and his colleagues on stochastic differential equations containing delays.