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

Raluca M. Balan

ESAIM: Probability and Statistics (2011)

  • 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 coefficientsf 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 (2011): 110-138. <http://eudml.org/doc/277158>.

@article{Balan2011,
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 coefficientsf and gk, driven by a sequence (βk)k of i.i.d. fractional Brownian motions of index H&gt;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},
pages = {110-138},
publisher = {EDP-Sciences},
title = {Lp-theory for the stochastic heat equation with infinite-dimensional fractional noise},
url = {http://eudml.org/doc/277158},
volume = {15},
year = {2011},
}

TY - JOUR
AU - Balan, Raluca M.
TI - Lp-theory for the stochastic heat equation with infinite-dimensional fractional noise
JO - ESAIM: Probability and Statistics
PY - 2011
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 coefficientsf and gk, driven by a sequence (βk)k of i.i.d. fractional Brownian motions of index H&gt;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/277158
ER -

References

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