Initial measures for the stochastic heat equation

Daniel Conus; Mathew Joseph; Davar Khoshnevisan; Shang-Yuan Shiu

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 1, page 136-153
  • ISSN: 0246-0203

Abstract

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We consider a family of nonlinear stochastic heat equations of the form ∂ t u = ℒ u + σ ( u ) W ˙ , where W ˙ denotes space–time white noise, ℒ the generator of a symmetric Lévy process on 𝐑 , and σ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure u 0 . Tight a priori bounds on the moments of the solution are also obtained. In the particular case that ℒ f = c f ' ' for some c g t ; 0 , we prove that if u 0 is a finite measure of compact support, then the solution is with probability one a bounded function for all times t g t ; 0 .

How to cite

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Conus, Daniel, et al. "Initial measures for the stochastic heat equation." Annales de l'I.H.P. Probabilités et statistiques 50.1 (2014): 136-153. <http://eudml.org/doc/272019>.

@article{Conus2014,
abstract = {We consider a family of nonlinear stochastic heat equations of the form $\partial _\{t\}u=\mathcal \{L\}u+\sigma (u)\dot\{W\}$, where $\dot\{W\}$ denotes space–time white noise, $\mathcal \{L\}$ the generator of a symmetric Lévy process on $\mathbf \{R\} $, and $\sigma $ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_\{0\}$. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $\mathcal \{L\}f=cf^\{\prime \prime \}$ for some $c&gt;0$, we prove that if $u_\{0\}$ is a finite measure of compact support, then the solution is with probability one a bounded function for all times $t&gt;0$.},
author = {Conus, Daniel, Joseph, Mathew, Khoshnevisan, Davar, Shiu, Shang-Yuan},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {The stochastic heat equation; singular initial data; stochastic heat equation},
language = {eng},
number = {1},
pages = {136-153},
publisher = {Gauthier-Villars},
title = {Initial measures for the stochastic heat equation},
url = {http://eudml.org/doc/272019},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Conus, Daniel
AU - Joseph, Mathew
AU - Khoshnevisan, Davar
AU - Shiu, Shang-Yuan
TI - Initial measures for the stochastic heat equation
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 1
SP - 136
EP - 153
AB - We consider a family of nonlinear stochastic heat equations of the form $\partial _{t}u=\mathcal {L}u+\sigma (u)\dot{W}$, where $\dot{W}$ denotes space–time white noise, $\mathcal {L}$ the generator of a symmetric Lévy process on $\mathbf {R} $, and $\sigma $ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_{0}$. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $\mathcal {L}f=cf^{\prime \prime }$ for some $c&gt;0$, we prove that if $u_{0}$ is a finite measure of compact support, then the solution is with probability one a bounded function for all times $t&gt;0$.
LA - eng
KW - The stochastic heat equation; singular initial data; stochastic heat equation
UR - http://eudml.org/doc/272019
ER -

References

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