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Intermittency properties in a hyperbolic Anderson problem

Robert C. Dalang; Carl Mueller

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 4, page 1150-1164
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Dalang, Robert C., and Mueller, Carl. "Intermittency properties in a hyperbolic Anderson problem." Annales de l'I.H.P. Probabilités et statistiques 45.4 (2009): 1150-1164. <http://eudml.org/doc/78058>.

@article{Dalang2009,
abstract = {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.},
author = {Dalang, Robert C., Mueller, Carl},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic wave equation; stochastic partial differential equations; moment Lyapunov exponents; intermittency; stochastic heat equation},
language = {eng},
number = {4},
pages = {1150-1164},
publisher = {Gauthier-Villars},
title = {Intermittency properties in a hyperbolic Anderson problem},
url = {http://eudml.org/doc/78058},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Dalang, Robert C.
AU - Mueller, Carl
TI - Intermittency properties in a hyperbolic Anderson problem
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 4
SP - 1150
EP - 1164
AB - 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.
LA - eng
KW - stochastic wave equation; stochastic partial differential equations; moment Lyapunov exponents; intermittency; stochastic heat equation
UR - http://eudml.org/doc/78058
ER -

References

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  1. [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. Zbl0972.60050MR1910468
  2. [2] R. A. Carmona and S. A. Molchanov. Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 (1994) 1–125. Zbl0925.35074MR1185878
  3. [3] D. Conus and R. C. Dalang. The non-linear stochastic wave equation in high dimensions. Electron. J. Probab. 13 (2008) 629–670. Zbl1187.60049MR2399293
  4. [4] M. Cranston and S. Molchanov. Quenched to annealed transition in the parabolic Anderson problem. Probab. Theory Related Fields 138 (2007) 177–193. Zbl1136.60016MR2288068
  5. [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. Zbl1082.60057MR2197105
  6. [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). Zbl0922.60056MR1684157
  7. [7] R. C. Dalang and N. E. Frangos. The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 (1998) 187–212. Zbl0938.60046MR1617046
  8. [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. Zbl1149.60040MR2403701
  9. [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. Zbl1214.60028MR2512755
  10. [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. Zbl0972.60056MR1808375
  11. [11] J. Gärtner, W. König and S. Molchanov. Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35 (2007) 439–499. Zbl1126.60091MR2308585
  12. [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. Zbl0814.60093MR1307415
  13. [13] L. Schwartz. Théorie des distributions. Hermann, Paris, 1966. Zbl0149.09501MR209834
  14. [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. MR989562
  15. [15] S. Tindel and F. Viens. Almost-sure exponential behaviour for a parabolic SPDE on a manifold. Stochastic Process. Appl. 100 (2002) 53–74. Zbl1058.60053MR1919608
  16. [16] J. B. Walsh. An introduction to stochastic partial differential equations. In Ecole d’été de probabilités de Saint-Flour, XIV—1984265–439. Lecture Notes in Math. 1180. Springer, Berlin, 1986. Zbl0608.60060MR876085
  17. [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. MR907831
  18. [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. 

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