Exponential mixing in a hydrodynamic pilot--wave theory with singular potentials

TL;DR

This paper proves that a stochastic hydrodynamic pilot-wave model with singular potentials exhibits exponential mixing, demonstrating ergodic behavior and convergence to a unique invariant measure, supported by theoretical analysis and numerical simulations.

Contribution

It extends ergodicity results to pilot-wave systems with singular potentials, using Lyapunov functions and asymptotic coupling techniques.

Findings

Walker dynamics are exponentially attracted to a unique invariant measure.

Numerical simulations illustrate invariant measures in Coulomb potentials.

Theoretical methods confirm ergodicity in systems with singularities.

Abstract

We conduct an analysis of a stochastic hydrodynamic pilot-wave theory, which is a Langevin equation with a memory kernel that describes the dynamics of a walking droplet (or "walker") subjected to a repulsive singular potential and random perturbations through additive Gaussian noise. Under suitable assumptions on the singularities, we show that the walker dynamics is exponentially attracted toward the unique invariant probability measure. The proof relies on a combination of the Lyapunov technique and an asymptotic coupling specifically tailored to our setting. We also present examples of invariant measures, as obtained from numerical simulations of the walker in two-dimensional Coulomb potentials. Our results extend previous work on the ergodicity of stochastic pilot-wave dynamics established for smooth confining potentials.