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.

Classification:  60H15,  37H15,  35L05
Keywords: stochastic wave equation, stochastic partial differential equations, moment Lyapunov exponents, intermittency, stochastic heat equation
     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 = {}
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.

[1] R. A. Carmona, L. Koralov and S. A. Molchanov. Asymptotics for the almost sure Lyapunov exponent for the solution of the parabolic Anderson problem. Random Oper. Stochastic Equations 9 (2001) 77-86. | MR 1910468 | Zbl 0972.60050

[2] R. A. Carmona and S. A. Molchanov. Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 (1994) 1-125. | MR 1185878 | Zbl 0925.35074

[3] D. Conus and R. C. Dalang. The non-linear stochastic wave equation in high dimensions. Electron. J. Probab. 13 (2008) 629-670. | MR 2399293 | Zbl pre05636517

[4] M. Cranston and S. Molchanov. Quenched to annealed transition in the parabolic Anderson problem. Probab. Theory Related Fields 138 (2007) 177-193. | MR 2288068 | Zbl 1136.60016

[5] M. Cranston, T. S. Mountford and T. Shiga. Lyapunov exponent for the parabolic Anderson model with Lévy noise. Probab. Theory Related Fields 132 (2005) 321-355. | MR 2197105 | Zbl 1082.60057

[6] R. C. Dalang. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s. Electron. J. Probab. 4 (1999) 1-29 (with Correction). | MR 1684157 | Zbl 0922.60056

[7] R. C. Dalang and N. E. Frangos. The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 (1998) 187-212. | MR 1617046 | Zbl 0938.60046

[8] R. C. Dalang, C. Mueller and R. Tribe. A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.'s. Trans. Amer. Math. Soc. 360 (2008) 4681-4703. | MR 2403701 | Zbl 1149.60040

[9] R. C. Dalang and M. Sanz-Solé. Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension 3. Mem. Amer. Math. Soc. (2009). To appear. | MR 2512755 | Zbl pre05577573

[10] J. Gärtner, W. König and S. A. Molchanov. Almost sure asymptotics for the continuous parabolic Anderson model. Probab. Theory Related Fields 118 (2000) 547-573. | MR 1808375 | Zbl 0972.60056

[11] J. Gärtner, W. König and S. Molchanov. Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35 (2007) 439-499. | MR 2308585 | Zbl 1126.60091

[12] S. Molchanov. Lectures on random media. In Lectures on Probability Theory (Saint-Flour, 1992) 242-411. Lecture Notes in Math. 1581. Springer, Berlin, 1994. | MR 1307415 | Zbl 0797.00021 | Zbl 0814.60093

[13] L. Schwartz. Théorie des distributions. Hermann, Paris, 1966. | MR 209834 | Zbl 0149.09501

[14] S. F. Shandarin and Ya. B. Zel'Dovich. The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium. Rev. Modern Phys. 61 (1989) 185-220. | MR 989562

[15] S. Tindel and F. Viens. Almost-sure exponential behaviour for a parabolic SPDE on a manifold. Stochastic Process. Appl. 100 (2002) 53-74. | MR 1919608 | Zbl 1058.60053

[16] J. B. Walsh. An introduction to stochastic partial differential equations. In Ecole d'été de probabilités de Saint-Flour, XIV-1984 265-439. Lecture Notes in Math. 1180. Springer, Berlin, 1986. | MR 876085 | Zbl 0608.60060

[17] Ya. B. Zel'Dovich, S. A. Molchanov, A. A. Ruzmaĭkin and D. D. Sokoloff. Self-excitation of a nonlinear scalar field in a random medium. Proc. Natl. Acad. Sci. U.S.A. 84 (1987) 6323-6325. | MR 907831

[18] Ya. B. Zel'Dovich, S. A. Molchanov, A. A. Ruzmaĭkin and D. D. Sokolov. Intermittency in random media. Uspekhi Fiz. Nauk 152 (1987) 3-32.