The Inert Drift Atlas Model

TL;DR

This paper analyzes a complex stochastic system modeling an inert particle influenced by Brownian particles and external potential, establishing its unique stationary distribution and exponential convergence rate.

Contribution

It introduces a novel analysis of a non-hypoelliptic, non-reversible system with singular interactions, deriving explicit stationary distribution and convergence properties.

Findings

Unique stationary distribution with Gaussian velocity and exponential gaps.

Exponential rate of convergence to stationarity.

Law of large numbers for particle locations and local times.

Abstract

Consider a massive (inert) particle impinged from above by N Brownian particles that are instantaneously reflected upon collision with the inert particle. The velocity of the inert particle increases due to the influence of an external Newtonian potential (e.g. gravitation) and decreases in proportion to the total local time of collisions with the Brownian particles. This system models a semi-permeable membrane in a fluid having microscopic impurities (Knight (2001)). We study the long-time behavior of the process $(V,\mathbf{Z})$, where $V$ is the velocity of the inert particle and $\mathbf{Z}$ is the vector of gaps between successive particles ordered by their relative positions. The system is not hypoelliptic, not reversible, and has singular form interactions. Thus the study of stability behavior of the system requires new ideas. We show that this process has a unique stationary distribution that takes an explicit product form which is Gaussian in the velocity component and Exponential in the other components. We also show that convergence in total variation distance to the stationary distribution happens at an exponential rate. We further obtain certain law of large numbers results for the particle locations and intersection local times.