Initial measures for the stochastic heat equation

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

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 136-153 / Harvested from Numdam

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 $\mathcal {L}f=cf^{\prime \prime }$ 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 $t>0$.

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 $c>0$, 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 $t>0$.

Détail

Publié le : 2014-01-01

MR 3161526 | Zbl 1288.60077

DOI : https://doi.org/10.1214/12-AIHP505
Classification:  60H15,  35R60

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