$L_p$-Theory for the Stochastic Heat Equation with Infinite-Dimensional Fractional Noise

Abstract

In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x))dt+ \sum_{k=1}^{\infty} g^{k}(t,x) \delta \beta_t^k, t \in [0,T]$, with random coefficients $f$ and $g^k$, driven by a sequence $(\beta^k)_k$ of i.i.d. fractional Brownian motions of index $H>1/2$. Using the Malliavin calculus techniques and a $p$-th moment maximal inequality for the infinite sum of Skorohod integrals with respect to $(\beta^k)_k$, we prove that the equation has a unique solution (in a Banach space of summability exponent $p \geq 2$), and this solution is H\"older continuous in both time and space.