# Inert drift system in a viscous fluid: Steady state asymptotics and exponential ergodicity

## TL;DR

This paper investigates the long-term behavior of a stochastic system modeling an inert particle in a viscous fluid with collisions, demonstrating exponential convergence to equilibrium and characterizing the stationary distribution's tails.

## Contribution

It introduces a renewal structure approach to analyze the system's ergodicity and provides explicit bounds on the stationary distribution's tails based on system parameters.

## Key findings

- Exponential convergence in total variation to the stationary distribution.
- Explicit bounds on the tails of the stationary distribution.
- Development of a renewal structure as a key analytical tool.

## Abstract

We analyze a system of stochastic differential equations describing the joint motion of a massive (inert) particle in a viscous fluid in the presence of a gravitational field and a Brownian particle impinging on it from below, which transfers momentum proportional to the local time of collisions. We study the long-time fluctuations of the velocity of the inert particle and the gap between the two particles, and we show convergence in total variation to the stationary distribution is exponentially fast. We also produce matching upper and lower bounds on the tails of the stationary distribution and show how these bounds depend on the system parameters. A renewal structure for the process is established, which is the key technical tool in proving the mentioned results.