$L_p$-theory for the stochastic heat equation with infinite-dimensional fractional noise

Balan, Raluca M.

ESAIM: Probability and Statistics, Tome 15 (2011), pp. 110-138.

In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x))\mathrm{d}t+{\sum }_{k=1}^{\infty }{g}^k(t,x)\delta {\beta }_t^k,t\in [0,T]$, with random coefficients f and g^k, driven by a sequence (β^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 (β^k)_k, we prove that the equation has a unique solution (in a Banach space of summability exponent p ≥ 2), and this solution is Hölder continuous in both time and space.

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