Central limit theorems for a driven particle in a random medium with mass aggregation

TL;DR

This work advances the stochastic analysis of a driven tracer in a random, mass-aggregating medium by proving central limit theorems for both the tracer’s position and velocity. Building on a prior LLN result for the limiting velocity, the authors first establish CLTs for a modified no-recollision process and then transfer these results to the original dynamics by showing the discrepancies are negligible at the diffusion scale. The limiting variances are connected to the auxiliary model via $\sigma_q^2$ and $\sigma_v^2$, with the long-time velocity limit $V_L = \sqrt{F\mu/(2-p)}$. The methodology combines detailed probabilistic decompositions, Lindeberg-Feller CLTs, and careful control of recollisions, contributing a rigorous fluctuation theory for mass-aggregation driven systems. These results are relevant for understanding emergent diffusion-like behavior in driven particle systems with random media and mass exchange.

Abstract

We establish central limit theorems for the position and velocity of the charged particle in the mechanical particle model introduced in the paper "Limit velocity for a driven particle in a random medium with mass aggregation" (https://doi.org/10.1016/S0246-0203(00)01059-1).

Theorems & Definitions (23)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 3.1
• proof
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• proof