Superdiffusivity for Brownian Motion in a Poissonian Potential with Long Range Correlation I: Lower Bound on the Volume Exponent

TL;DR

This paper investigates the superdiffusive behavior of Brownian motion in a long-range correlated Poissonian potential, establishing a lower bound on the volume exponent and highlighting the impact of correlations on superdiffusivity.

Contribution

It provides the first lower bound on the volume exponent for Brownian motion in a long-range correlated Poissonian potential with power-law trap radii.

Findings

Superdiffusivity holds under certain conditions.

Correlation enhances superdiffusivity.

Lower bound on the volume exponent established.

Abstract

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 can be associated to 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 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 W\"uhtrich: the superdiffusivity phenomenon is enhanced by the presence of correlation.