At the edge of a cloud of Brownian particles

TL;DR

This paper models a Brownian particle in a turbulent fluid with a Gaussian velocity field, showing that fluctuations near the weak-strong disorder transition are described by the KPZ equation, suggesting a universal behavior.

Contribution

It introduces a model for particle trajectories in turbulent fluids and links the transition fluctuations to the KPZ equation, extending understanding of turbulence and stochastic processes.

Findings

Fluctuations at the weak-strong disorder transition follow the KPZ equation.

The model connects turbulence with stochastic growth processes.

Conjecture extends results beyond weak environment assumptions.

Abstract

We study a simple model for the trajectory of a particle in a turbulent fluid, where a Brownian motion travels through a random Gaussian velocity field. We study the quenched law of the process and prove that in a weak environment setting, the fluctuations at the transition from weak to strong disorder are described by the KPZ equation. We conjecture the same is true without the assumption of a weak environment.