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

2009/05/13 by Raluca M. Balan, Raluca Balan, Balan, Raluca
Economics, Econometrics and Finance · Mathematics · #Financial Risk and Volatility Modeling #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60H07 #msc:60H15

paper · pdf · doi:10.48550/arxiv.0905.2150

arxiv created 2009/05/13 · arxiv updated 2009/12/01

Abstract

In this article, we consider the stochastic heat equation du=(Δu+f(t,x))dt+ ∑_k=1^∞ g^k(t,x) δβ_t^k, t ∈ [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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