Initial measures for the stochastic heat equation
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, p. 136-153

We consider a family of nonlinear stochastic heat equations of the form t u=u+σ(u)W ˙, where W ˙ denotes space-time white noise, 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 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.

Nous considérons une famille d’équations de la chaleur stochastique de la forme t u=u+σ(u)W ˙, où W ˙ est un bruit-blanc espace-temps, est le générateur d’un processus de Lévy symétrique sur 𝐑, et σ 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 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.

DOI : https://doi.org/10.1214/12-AIHP505
Classification:  60H15,  35R60
Keywords: The stochastic heat equation, singular initial data
@article{AIHPB_2014__50_1_136_0,
     author = {Conus, Daniel and Joseph, Mathew and Khoshnevisan, Davar and Shiu, Shang-Yuan},
     title = {Initial measures for the stochastic heat equation},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {1},
     year = {2014},
     pages = {136-153},
     doi = {10.1214/12-AIHP505},
     zbl = {1288.60077},
     mrnumber = {3161526},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_1_136_0}
}
Conus, Daniel; Joseph, Mathew; Khoshnevisan, Davar; Shiu, Shang-Yuan. Initial measures for the stochastic heat equation. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 136-153. doi : 10.1214/12-AIHP505. http://www.numdam.org/item/AIHPB_2014__50_1_136_0/

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