Brownian motion and parabolic Anderson model in a renormalized Poisson potential
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 3, pp. 631-660.

Nous présentons une méthode de renormalisation pour construire certains modèles de potentiels aléatoires dans un nuage Poissonnien qui sont physiquement plus réalistes. Nous obtenons le mouvement brownien dans un potentiel aléatoire renormalisé et les modèles d'Anderson paraboliques associés. Par exemple, avec cette renormalisation, nous pouvons construire rigoureusement des modèles consistants avec la loi de la gravitation de Newton.

A method known as renormalization is proposed for constructing some more physically realistic random potentials in a Poisson cloud. The Brownian motion in the renormalized random potential and related parabolic Anderson models are modeled. With the renormalization, for example, the models consistent to Newton's law of universal attraction can be rigorously constructed.

DOI : 10.1214/11-AIHP419
Classification : 60J45, 60J65, 60K37, 60K37, 60G55
Mots clés : renormalization, Poisson field, brownian motion in Poisson potential, parabolic Anderson model, Newton's law of universal attraction
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     title = {Brownian motion and parabolic {Anderson} model in a renormalized {Poisson} potential},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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Chen, Xia; Kulik, Alexey M. Brownian motion and parabolic Anderson model in a renormalized Poisson potential. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 3, pp. 631-660. doi : 10.1214/11-AIHP419. http://archive.numdam.org/articles/10.1214/11-AIHP419/

[1] R. Bass, X. Chen and J. Rosen. Large deviations for Riesz potentials of additive processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 626-666. | Numdam | MR | Zbl

[2] S. Bezerra, S. Tindel and F. Viens. Superdiffusivity for a Brownian polymer in a continuous Gaussian environment. Ann. Probab. 36 (2008) 1642-1675. | MR | Zbl

[3] M. Biskup and W. König. Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model. J. Statist. Phys. 102 (2001) 1253-1270. | MR | Zbl

[4] V. S. Borkar. Probability Theory: An Advanced Course. Springer, New York, 1995. | MR | Zbl

[5] R. A. Carmona and S. A. Molchanov. Parabolic Anderson Problem and Intermittency. Amer. Math. Soc., Providence, RI, 1994. | MR | Zbl

[6] R. A. Carmona and F. G. Viens. Almost-sure exponential behavior of a stochastic Anderson model with continuous space parameter. Stochastics 62 (1998) 251-273. | MR | Zbl

[7] X. Chen. Random Walk Intersections: Large Deviations and Related Topics. Math. Surv. Mono. 157. Amer. Math. Soc., Providence, RI, 2009. | MR | Zbl

[8] X. Chen. Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related Anderson models. Ann. Probab. 40 (2012) 1436-1482. | MR | Zbl

[9] X. Chen and A. M. Kulik. Asymptotics of negative exponential moments for annealed Brownian motion in a renormalized Poisson potential. Preprint, 2011. | MR | Zbl

[10] X. Chen and J. Rosinski. Spatial Brownian motion in renormalized Poisson potential: A critical case. Preprint, 2011.

[11] M. Cranston, D. Gauthier and T. S. Mountford. On large deviations for the parabolic Anderson model. Probab. Theory Related Fields 147 (2010) 349-378. | MR | Zbl

[12] R. C. Dalang and C. Mueller. Intermittency properties in a hyperbolic Anderson problem. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 1150-1164. | Numdam | MR | Zbl

[13] A. De Acosta. Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm. Ann. Probab. 11 (1983) 78-101. | MR | Zbl

[14] M. D. Donsker and S. R. S. Varadhan. Asymptotics for the Wiener sausage. Comm. Pure Appl. Math. 28 (1975) 525-565. | MR | Zbl

[15] I. Florescu and F. Viens. Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space. Probab. Theory Related Fields 135 (2006) 603-644. | Zbl

[16] R. Fukushima. Second order asymptotics for Brownian motion among a heavy tailed Poissonian potential. Preprint, 2010. | Zbl

[17] J. Gärtner, F. Den Hollander and G. Maillard. Intermittency on catalysts: Symmetric exclusion. Electron. J. Probab. 12 (2007) 516-573. | Zbl

[18] J. Gärtner and W. König. Moment asymptotics for the continuous parabolic Anderson model. Ann. Appl. Probab. 10 (2000) 192-217. | Zbl

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

[20] J. Gärtner and S. A. Molchanov. Parabolic problem for the Anderson model. Comm. Math. Phys. 132 (1990) 613-655. | Zbl

[21] F. Germinet, P. Hislop and A. Klein. Localization for Schrödinger operators with Poisson random potential. J. Europ. Math. Soc. 9 (2007) 577-607. | MR | Zbl

[22] S. Harvlin and D. Ben Avraham. Diffusion in disordered media. Adv. in Phys. 36 (1987) 695-798.

[23] T. Komorowski. Brownian motion in a Poisson obstacle field. Séminaire Bourbaki 1998/99 (2000) 91-111. | Numdam | MR | Zbl

[24] M. B. Marcus and J. Rosinski. Continuity and boundedness of infinitely divisible process: A Poisson point process approach. J. Theoret. Probab. 18 (2005) 109-160. | MR | Zbl

[25] L. A. Pastur. The behavior of certain Wiener integrals as t and the density of states of Schrödinger equations with random potential. Teoret. Mat. Fiz. 32 (1977) 88-95. | MR | Zbl

[26] A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin, 1983. | MR | Zbl

[27] T. Povel. Confinement of Brownian motion among Poissonian obstacles in d , d3. Probab. Theory Related Fields 114 (1999) 177-205. | MR | Zbl

[28] B. S. Rajput and J. Rosinski. Spectral representations of infinitely divisible processes. Probab. Theory Related Fields 82 (1989) 451-487. | MR | Zbl

[29] J. Rosinski. On path properties of certain infinitely divisible process. Stochastic Process. Appl. 33 (1989) 73-87. | MR | Zbl

[30] G. Stolz. Non-monotonic random Schrödinger operators: The Anderson model. J. Math. Anal. Appl. 248 (2000) 173-183. | MR | Zbl

[31] A.-L. Sznitman. Brownian Motion, Obstacles and Random Media. Springer, Berlin, 1998. | MR | Zbl

[32] M. Van Den Berg, E. Bolthausen and F. Den Hollander. Brownian survival among Poissonian traps with random shapes at critical intensity. Probab. Theory Related Fields 132 (2005) 163-202. | MR | Zbl

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