Mild Solution of Semilinear Rough Stochastic Evolution Equations

In this paper, we investigate a semilinear stochastic parabolic equation with a linear rough term $d{u}_t=[L_t{u}_t+f(t,u_t)]dt+(G_t{u}_t+g_t)d{\mathbf{X}}_t+h(t,u_t)d{W}_t$, where ${\left(L_t\right)}_{t\in [0,T]}$ is a family of unbounded operators acting on a monotone family of interpolation Hilbert spaces, $\mathbf{X}$ is a two-step $\alpha$-Hölder rough path with $\alpha \in (1\mathrm{/}3,1\mathrm{/}2]$ and $W$ is a Brownian motion. Existence and uniqueness of the mild solution are given through the stochastic controlled rough path approach and fixed-point argument. As a technical tool to define rough stochastic convolutions, we also develop a general mild stochastic sewing lemma, which is applicable for processes according to a monotone family.