Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Initial measures for the stochastic heat equation

Daniel Conus, Mathew Joseph, Davar Khoshnevisan, and Shang-Yuan Shiu

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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 $\mathcal{L}f=cf''$ for some $c>0$, 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 $t>0$.

Résumé

Nous considérons une famille d’équations de la chaleur stochastique de la forme $\partial_{t}u=\mathcal{L}u+\sigma(u)\dot{W}$, où $\dot{W}$ est un bruit-blanc espace–temps, $\mathcal{L}$ est le générateur d’un processus de Lévy symétrique sur $\mathbf{R} $, et $\sigma$ est une fonction lipschizienne s’annulant en $0$. Nous montrons que cette équation aux dérivées partielles stochastique a une solution de type champ aléatoire pour toute mesure initiale finie $u_{0}$. Nous obtenons également des bornes a priori sur les moments de la solution.

Dans le cas particulier où $\mathcal{L}f=cf''$ pour un $c>0$, nous montrons que si $u_{0}$ est une mesure finie à support compact, la solution est presque sûrement une fonction bornée pour tout $t>0$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 1 (2014), 136-153.

Dates
First available in Project Euclid: 1 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1388545269

Digital Object Identifier
doi:10.1214/12-AIHP505

Mathematical Reviews number (MathSciNet)
MR3161526

Zentralblatt MATH identifier
1288.60077

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Keywords
The stochastic heat equation Singular initial data

Citation

Conus, Daniel; Joseph, Mathew; Khoshnevisan, Davar; Shiu, Shang-Yuan. Initial measures for the stochastic heat equation. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 1, 136--153. doi:10.1214/12-AIHP505. https://projecteuclid.org/euclid.aihp/1388545269


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