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

Raluca M. Balan

ESAIM: Probability and Statistics (2012)

  • Volume: 15, page 110-138
  • ISSN: 1292-8100

Abstract

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In this article, we consider the stochastic heat equation d u = ( Δ u + f ( t , x ) ) d t + k = 1 g k ( t , x ) δ β t k , t [ 0 , T ] , with random coefficients f and gk, driven by a sequence (βk)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 (βk)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.

How to cite

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Balan, Raluca M.. "Lp-theory for the stochastic heat equation with infinite-dimensional fractional noise*." ESAIM: Probability and Statistics 15 (2012): 110-138. <http://eudml.org/doc/222479>.

@article{Balan2012,
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 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 (βk)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. },
author = {Balan, Raluca M.},
journal = {ESAIM: Probability and Statistics},
keywords = {Fractional Brownian motion; Skorohod integral; maximal inequality; stochastic heat equation; fractional Brownian motion},
language = {eng},
month = {1},
pages = {110-138},
publisher = {EDP Sciences},
title = {Lp-theory for the stochastic heat equation with infinite-dimensional fractional noise*},
url = {http://eudml.org/doc/222479},
volume = {15},
year = {2012},
}

TY - JOUR
AU - Balan, Raluca M.
TI - Lp-theory for the stochastic heat equation with infinite-dimensional fractional noise*
JO - ESAIM: Probability and Statistics
DA - 2012/1//
PB - EDP Sciences
VL - 15
SP - 110
EP - 138
AB - 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 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 (βk)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.
LA - eng
KW - Fractional Brownian motion; Skorohod integral; maximal inequality; stochastic heat equation; fractional Brownian motion
UR - http://eudml.org/doc/222479
ER -

References

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