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

TL;DR

This paper develops an $L_p$-theory for the stochastic heat equation driven by infinite-dimensional fractional noise, establishing existence, uniqueness, and regularity of solutions using Malliavin calculus techniques.

Contribution

It introduces a novel $L_p$-framework for the stochastic heat equation with infinite-dimensional fractional noise, proving well-posedness and regularity results.

Findings

Unique solution exists in a Banach space for $p \\geq 2$

Solution exhibits Hölder continuity in time and space

Maximal inequality for infinite Skorohod integrals established

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.