Superdiffusivity for brownian motion in a poissonian potential with long range correlation II: Upper bound on the volume exponent

Lacoin, Hubert

doi:10.1214/11-AIHP457

Lacoin, Hubert

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 1029-1048.

Résumé
Abstract

Cet article est la seconde partie d’une étude sur les trajectoires Brownienne dans un champs de pièges mous dont le rayon est aléatoire et a une distribution non-bornée. Nous montrons que l’exposant de volume (qui est l’exposant associé aux fluctuations transversales des trajectoires) $ξ$ est strictement inférieur à $1$ et nous donnons une borne supérieure explicite qui dépend des paramètres du problème, et ceci aussi bien pour le modèle dans la configuration point-à-point que pour celui dans la configuration point à plan. Dans certains cas particulier, cette borne supérieure coïncide avec la borne inférieure démontrée dans la première partie de cette étude, ce qui nous permets d’identifier la valeur de l’exposant de volume.

This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here that for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) $ξ$ is strictly less than $1$ and give an explicit upper bound that depends on the parameters of the problem. In some specific cases, this upper bound matches the lower bound proved in the first part of this work and we get the exact value of the volume exponent.

MR Zbl | 2 citations dans Numdam

DOI : 10.1214/11-AIHP457

Classification : 82D60, 60K37, 82B44
Mots-clés : stretched polymer, quenched disorder, superdiffusivity, brownian motion, poissonian obstacles, correlation

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     title = {Superdiffusivity for brownian motion in a poissonian potential with long range correlation {II:} {Upper} bound on the volume exponent},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1029--1048},
     publisher = {Gauthier-Villars},
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     year = {2012},
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     mrnumber = {3052457},
     zbl = {1267.82147},
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     url = {http://archive.numdam.org/articles/10.1214/11-AIHP457/}
}

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Lacoin, Hubert. Superdiffusivity for brownian motion in a poissonian potential with long range correlation II: Upper bound on the volume exponent. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 1029-1048. doi : 10.1214/11-AIHP457. http://archive.numdam.org/articles/10.1214/11-AIHP457/

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