Intermittency properties in a hyperbolic Anderson problem
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 4, p. 1150-1164

We study the asymptotics of the even moments of solutions to a stochastic wave equation in spatial dimension 3 with linear multiplicative spatially homogeneous gaussian noise that is white in time. Our main theorem states that these moments grow more quickly than one might expect. This phenomenon is well known for parabolic stochastic partial differential equations, under the name of intermittency. Our results seem to be the first example of this phenomenon for hyperbolic equations. For comparison, we also derive bounds on moments of the solution to the stochastic heat equation with the same linear multiplicative noise.

Nous étudions le comportement asymptotique des moments pairs de la solution d'une équation des ondes stochastique en dimension spatiale 3 avec bruit gaussien multiplicatif linéaire spatiallement homogène et blanc en temps. Notre résultat principal affirme que ces moments croissent plus rapidement qu'attendu. Ce phénomène est bien connu dans le cadre d'équations aux dérivées partielles stochastiques paraboliques, sous le nom d' «intermittence.» Nos résultats mettent en évidence ce phénomène pour la première fois dans le cadre d'équations hyperboliques. Afin de comparer les deux situations, nous établissons aussi des bornes sur les moments de la solution d'une équation de la chaleur stochastique avec le même bruit multiplicatif linéaire.

DOI : https://doi.org/10.1214/08-AIHP199
Classification:  60H15,  37H15,  35L05
Keywords: stochastic wave equation, stochastic partial differential equations, moment Lyapunov exponents, intermittency, stochastic heat equation
@article{AIHPB_2009__45_4_1150_0,
     author = {Dalang, Robert C. and Mueller, Carl},
     title = {Intermittency properties in a hyperbolic Anderson problem},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {4},
     year = {2009},
     pages = {1150-1164},
     doi = {10.1214/08-AIHP199},
     zbl = {1196.60116},
     mrnumber = {2572169},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_4_1150_0}
}
Dalang, Robert C.; Mueller, Carl. Intermittency properties in a hyperbolic Anderson problem. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 4, pp. 1150-1164. doi : 10.1214/08-AIHP199. http://www.numdam.org/item/AIHPB_2009__45_4_1150_0/

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