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Lp-theory for the stochastic heat equationwith infinite-dimensional fractional noise*

Published online by Cambridge University Press:  05 January 2012

Raluca M. Balan
Raluca M. Balan*
Affiliation:
University of Ottawa, Department of Mathematics and Statistics, 585 King Edward Avenue Ottawa, ON, K1N 6N5, Canada; rbalan@uottawa.ca, http://aix1.uottawa.ca/~rbalan
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Abstract

In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x)){\rm d}t+ \sum_{k=1}^{\infty} g^{k}(t,x) \delta \beta_t^k, t \in [0,T]$, with random coefficients f and gk,driven by a sequence (βk)k of i.i.d. fractional Brownianmotions of index H>1/2. Using the Malliavin calculus techniquesand a p-th moment maximal inequality for the infinite sum ofSkorohod integrals with respect to (βk)k, we prove that theequation has a unique solution (in a Banach space of summabilityexponent p ≥ 2), and this solution is Hölder continuous inboth time and space.

Keywords

Fractional Brownian motionSkorohod integralmaximal inequalitystochastic heat equation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 110 - 138
Copyright
© EDP Sciences, SMAI, 2011

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