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) no. 1, pp. 136-153.

Nous considérons une famille d’équations de la chaleur stochastique de la forme ${\partial }_{t}u=ℒu+\sigma \left(u\right)\dot{W}$, où $\dot{W}$ est un bruit-blanc espace-temps, $ℒ$ est le générateur d’un processus de Lévy symétrique sur $𝐑$, 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 $ℒf=c{f}^{\text{'}\text{'}}$ 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 $t>0$.

We consider a family of nonlinear stochastic heat equations of the form ${\partial }_{t}u=ℒu+\sigma \left(u\right)\dot{W}$, where $\dot{W}$ denotes space-time white noise, $ℒ$ the generator of a symmetric Lévy process on $𝐑$, 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 $ℒf=c{f}^{\text{'}\text{'}}$ 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 $t>0$.

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