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

Lacoin, Hubert

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 1010-1028.

Dans cet article, nous étudions les trajectoires d’un mouvement brownien dans ${ℝ}^d$ évoluant dans un potentiel poissonien jusqu’au temps d’atteinte d’un hyper-plan situé loin de l’origine. Le potentiel poissonien $V$ que nous considerons est construit à partir d’un champs de pièges dont les centres sont déterminés par un processus de Poisson et dont les rayons sont des variables aléatoires IID. Nous concentrons notre étude sur le cas particulier ou la loi des rayons des pièges à une queue polynomiale et nous prouvons que les trajectoires ont un caractère surdiffusif quand certaines conditions sont vérifées et nous donnons une borne inférieure pour l’exposant de volume. Les résultats sont sensiblement différents de ceux obtenus dans le cas ou les pièges sont à rayon bornés par Wühtrich (Ann. Probab. 26 (1998) 1000-1015, Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 279-308) : le phénomène de surdiffusivité est renforcé par la présence de corrélations.

We study trajectories of $d$-dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential $V$ is constructed from a field of traps whose centers location is given by a Poisson Point Process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii $\nu$ has power-law decay and prove that superdiffusivity hold under certain condition, and get a lower bound on the volume exponent. Results differ quite much with the one that have been obtained for the model with traps of bounded radii by Wühtrich (Ann. Probab. 26 (1998) 1000-1015, Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 279-308): the superdiffusivity phenomenon is enhanced by the presence of correlation.

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