We consider a family of nonlinear stochastic heat equations of the form , where denotes space-time white noise, the generator of a symmetric Lévy process on , and is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure . Tight a priori bounds on the moments of the solution are also obtained. In the particular case that for some , we prove that if is a finite measure of compact support, then the solution is with probability one a bounded function for all times .
Nous considérons une famille d’équations de la chaleur stochastique de la forme , où est un bruit-blanc espace-temps, est le générateur d’un processus de Lévy symétrique sur , et est une fonction lipschizienne s’annulant en . Nous montrons que cette équation aux dérivées partielles stochastique a une solution de type champ aléatoire pour toute mesure initiale finie . Nous obtenons également des bornes a priori sur les moments de la solution. Dans le cas particulier où pour un , nous montrons que si est une mesure finie à support compact, la solution est presque sûrement une fonction bornée pour tout .
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}, pages = {136--153}, publisher = {Gauthier-Villars}, volume = {50}, number = {1}, year = {2014}, doi = {10.1214/12-AIHP505}, mrnumber = {3161526}, zbl = {1288.60077}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/12-AIHP505/} }
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 SP - 136 EP - 153 VL - 50 IS - 1 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/12-AIHP505/ DO - 10.1214/12-AIHP505 LA - en ID - AIHPB_2014__50_1_136_0 ER -
%0 Journal Article %A Conus, Daniel %A Joseph, Mathew %A Khoshnevisan, Davar %A Shiu, Shang-Yuan %T Initial measures for the stochastic heat equation %J Annales de l'I.H.P. Probabilités et statistiques %D 2014 %P 136-153 %V 50 %N 1 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/12-AIHP505/ %R 10.1214/12-AIHP505 %G en %F 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://archive.numdam.org/articles/10.1214/12-AIHP505/
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