Gravitation versus Brownian motion

TL;DR

This paper models the interaction between a massive inert particle under gravity and a Brownian particle, analyzing their joint dynamics, fluctuations, and convergence to a stationary distribution with explicit form.

Contribution

It introduces a novel stochastic model combining Brownian motion and gravitational effects, providing explicit stationary distribution results.

Findings

Velocity and gap distributions converge to a known stationary distribution.

The joint distribution has an explicit product form.

Fluctuations and strong laws of the particles are characterized.

Abstract

We investigate the motion of an inert (massive) particle being impinged from below by a particle performing (reflected) Brownian motion. The velocity of the inert particle increases in proportion to the local time of collisions and decreases according to a constant downward gravitational acceleration. We study fluctuations and strong laws of the motion of the particles. We further show that the joint distribution of the velocity of the inert particle and the gap between the two particles converges in total variation distance to a stationary distribution which has an explicit product form.