Nous considérons le modèle Anderson parabolique donné par le problème de Cauchy pour l'équation de la chaleur avec un potentiel aléatoire dans ℤd. Nous utilisons des potentiels i.i.d. ξ:ℤd→ℝ qui sont dans la troisième classe d'universalité de la classification de van der Hofstad, König and Mörters [Commun. Math. Phys. 267 (2006) 307-353]. Cette classe, les potentiels presque bornés, contient les potentiels dont le logarithme de la transformée de Laplace est de variation régulière avec paramètre γ=1, mais qui n'appartiennent pas à la classe des potentiels «double-exponentially distributed» étudiée par Gärtner et Molchanov dans [Probab. Theory Related Fields 111 (1998) 17-55]. Dans [Commun. Math. Phys. 267 (2006) 307-353], le comportement asymptotique de l'espérance de la masse totale est décrit par un problème variationnel qui est lié à l'inégalité de Sobolev logarithmique. La solution de ce problème, qui est unique à des translations spatiales près, est une parabole. Dans cet article, nous montrons que la contribution à la masse totale des potentiels qui ont (après un changement d'échelle) une forme différente est négligeable. Nous utilisons la topologie L1 sur les compacts pour l'exponentielle du potentiel. Au cours de la preuve, nous montrons que n'importe quelle suite des solutions approximatives du problème variationnel converge vers une translation spatiale de la solution qui est une parabole.
We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in ℤd. We use i.i.d. potentials ξ:ℤd→ℝ in the third universality class, namely the class of almost bounded potentials, in the classification of van der Hofstad, König and Mörters [Commun. Math. Phys. 267 (2006) 307-353]. This class consists of potentials whose logarithmic moment generating function is regularly varying with parameter γ=1, but do not belong to the class of so-called double-exponentially distributed potentials studied by Gärtner and Molchanov [Probab. Theory Related Fields 111 (1998) 17-55]. In [Commun. Math. Phys. 267 (2006) 307-353] the asymptotics of the expected total mass was identified in terms of a variational problem that is closely connected to the well-known logarithmic Sobolev inequality and whose solution, unique up to spatial shifts, is a perfect parabola. In the present paper we show that those potentials whose shape (after appropriate vertical shifting and spatial rescaling) is away from that parabola contribute only negligibly to the total mass. The topology used is the strong L1-topology on compacts for the exponentials of the potential. In the course of the proof, we show that any sequence of approximate minimisers of the above variational formula approaches some spatial shift of the minimiser, the parabola.
Mots clés : parabolic Anderson problem, intermittency, logarithmic Sobolev inequality, potential shape, Feynman-Kac formula
@article{AIHPB_2009__45_3_840_0, author = {Gr\"uninger, Gabriela and K\"onig, Wolfgang}, title = {Potential confinement property of the parabolic {Anderson} model}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {840--863}, publisher = {Gauthier-Villars}, volume = {45}, number = {3}, year = {2009}, doi = {10.1214/08-AIHP197}, mrnumber = {2548507}, zbl = {1185.60073}, language = {en}, url = {http://www.numdam.org/articles/10.1214/08-AIHP197/} }
TY - JOUR AU - Grüninger, Gabriela AU - König, Wolfgang TI - Potential confinement property of the parabolic Anderson model JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 840 EP - 863 VL - 45 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/08-AIHP197/ DO - 10.1214/08-AIHP197 LA - en ID - AIHPB_2009__45_3_840_0 ER -
%0 Journal Article %A Grüninger, Gabriela %A König, Wolfgang %T Potential confinement property of the parabolic Anderson model %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 840-863 %V 45 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/08-AIHP197/ %R 10.1214/08-AIHP197 %G en %F AIHPB_2009__45_3_840_0
Grüninger, Gabriela; König, Wolfgang. Potential confinement property of the parabolic Anderson model. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 840-863. doi : 10.1214/08-AIHP197. http://www.numdam.org/articles/10.1214/08-AIHP197/
[1] Regular Variation. Cambridge Univ. Press, 1987. | MR | Zbl
, and .[2] Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29 (2001) 636-682. | MR | Zbl
and .[3] Localization of a two-dimensional random walk with an attractive path interaction. Ann. Probab. 22 (1994) 875-918. | MR | Zbl
.[4] Superadditivity of Fisher's information and logarithmic Sobolev inequality. J. Funct. Anal. 101 (1991) 194-211. | MR | Zbl
.[5] Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 (1994) 1-125. | MR | Zbl
and .[6] Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. | MR | Zbl
and .[7] Correlation structure of intermittency in the parabolic Anderson model. Probab. Theory Related Fields 114 (1999) 1-54. | MR | Zbl
and .[8] The parabolic Anderson model. In Interacting Stochastic Systems 153-179. J.-D. Deuschel and A. Greven (Eds). Springer, Berlin, 2005. | MR | Zbl
and .[9] Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35 (2007) 439-499. | MR | Zbl
, and .[10] Parabolic problems for the Anderson model I. Intermittency and related topics. Commun. Math. Phys. 132 (1990) 613-655. | MR | Zbl
and .[11] Parabolic problems for the Anderson model II. Second-order asymptotics and structure of high peaks. Probab. Theory Related Fields 111 (1998) 17-55. | MR | Zbl
and .[12] The universality classes in the parabolic Anderson model. Commun. Math. Phys. 267 (2006) 307-353. | Zbl
, and .[13] Analysis, 2nd edition. AMS Graduate Studies 14. Amer. Math. Soc., Providence, RI, 2001. | MR | Zbl
and .[14] Lectures on random media. In Lectures on Probability Theory. Ecole d'Eté de Probabilités de Saint-Flour XXII-1992 242-411. D. Bakry, R. D. Gill and S. Molchanov (Eds). Lecture Notes in Math. 1581. Springer, Berlin, 1994. | MR | Zbl
.[15] Confinement of Brownian motion among Poissonian obstacles in Rd, d≥3. Probab. Theory Related Fields 114 (1999) 177-205. | MR | Zbl
.[16] On the confinement property of two-dimensional Brownian motion among Poissonian obstacles. Comm. Pure Appl. Math. 44 (1991) 1137-1170. | MR | Zbl
.Cité par Sources :