$L_p$-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 $p\ge 2$), and this solution is Hölder continuous in both time and space.

References in corpus (2)

• The Stochastic Heat Equation with a Fractional-Colored Noise: Existence of the Solution
• Mild Solutions for a Class of Fractional SPDEs and Their Sample Paths