Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Hubert Lacoin

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Abstract

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) $\xi$ 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.

Résumé

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) $\xi$ 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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 48, Number 4 (2012), 1029-1048.

Dates
First available in Project Euclid: 16 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1353098439

Digital Object Identifier
doi:10.1214/11-AIHP457

Mathematical Reviews number (MathSciNet)
MR3052457

Zentralblatt MATH identifier
1267.82147

Subjects
Primary: 82D60: Polymers 60K37: Processes in random environments 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Stretched polymer Quenched disorder Superdiffusivity Brownian motion Poissonian obstacles Correlation

Citation

Lacoin, Hubert. Superdiffusivity for Brownian motion in a Poissonian potential with long range correlation II: Upper bound on the volume exponent. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 4, 1029--1048. doi:10.1214/11-AIHP457. https://projecteuclid.org/euclid.aihp/1353098439


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