Mild Solution of Semilinear Rough Stochastic Evolution Equations

Abstract

In this paper, we investigate a semilinear stochastic parabolic equation with a linear rough term $d{u}_t=\left[L_t{u}_t+f\left(t,u_t\right)\right]dt+\left(G_t{u}_t+g_t\right)d{\mathbf{X}}_t+h\left(t,u_t\right)d{W}_t$, where ${\left(L_t\right)}_{t\in \left[0,T\right]}$ is a family of unbounded operators acting on a monotone family of interpolation Hilbert spaces, $\mathbf{X}$ is a two-step $\hat{I}\pm$-Hölder rough path with $\hat{I}\pm \in \left(1/3,1/2\right]$ 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.

22 pages