Mild Solution of Semilinear SPDEs with Young Drifts

Abstract

In this paper, we study a semilinear SPDE with a linear Young drift $d{u}_t=L{u}_{t}dt+f\left(t,u_t\right)dt+\left(G_t{u}_t+g_t\right)d{\eta }_t+h\left(t,u_t\right)d{W}_t$, where $L$ is the generator of an analytical semigroup, $\eta$ is an $\alpha$-Hölder continuous path with $\alpha \in \left(1/2,1\right)$ and $W$ is a Brownian motion. After establishing through two different approaches the Young convolution integrals for stochastic integrands, we introduce the corresponding definition of mild solutions and continuous mild solutions, and give via a fixed-point argument the existence and uniqueness of the (continuous) mild solution under suitable conditions.

17 pages