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

TL;DR

This paper investigates the behavior of Brownian motion in a Poissonian potential with long-range correlations, establishing an explicit upper bound on the volume exponent that describes transversal fluctuations, and identifying cases where this bound is exact.

Contribution

It provides the first explicit upper bound on the volume exponent for Brownian motion in a correlated Poissonian potential, matching the lower bound in specific cases.

Findings

Volume exponent is strictly less than one.

Explicit upper bound depends on model parameters.

Exact volume exponent determined in special cases.

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 for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) is strictly less than one 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.