Long time asymptotics of a Brownian particle coupled with a random environment with non-diffusive feedback force

TL;DR

This paper investigates the long-term behavior of a Brownian particle in anomalously diffusing environments, revealing bounded variance in subdiffusive cases and polynomial growth in superdiffusive cases, using fractional calculus models.

Contribution

It introduces a detailed analysis of Brownian motion in non-diffusive fields with fractional operators, highlighting the different asymptotic behaviors based on environment type.

Findings

Bounded variance in subdiffusive environments.

Variance grows as t^{2γ-1} in superdiffusive environments.

Comparison of fractional Laplacian and Riemann-Liouville models.

Abstract

We study the long time behavior of a Brownian particle moving in an anomalously diffusing field, the evolution of which depends on the particle position. We prove that the process describing the asymptotic behaviour of the Brownian particle has bounded (in time) variance when the particle interacts with a subdiffusive field; when the interaction is with a superdiffusive field the variance of the limiting process grows in time as t^{2{\gamma}-1}, 1/2 < {\gamma} < 1. Two different kinds of superdiffusing (random) environments are considered: one is described through the use of the fractional Laplacian; the other via the Riemann-Liouville fractional integral. The subdiffusive field is modeled through the Riemann-Liouville fractional derivative.