The invariant measure of a walking droplet in hydrodynamic pilot-wave theory

TL;DR

This paper rigorously analyzes the long-term statistical behavior of a hydrodynamic pilot-wave system, establishing the existence and uniqueness of an invariant measure under certain conditions, supported by numerical simulations.

Contribution

It provides a rigorous mathematical framework for the long-time behavior of a walker in a pilot-wave system, including the construction of invariant measures.

Findings

Existence of a unique invariant measure under potential-dominated forces

Construction of solutions as dynamics on path spaces

Numerical example demonstrating the invariant measure in a harmonic potential

Abstract

We study the long time statistics of a walker in a hydrodynamic pilot-wave system, which is a stochastic Langevin dynamics with an external potential and memory kernel. While prior experiments and numerical simulations have indicated that the system may reach a statistically steady state, its long-time behavior has not been studied rigorously. For a broad class of external potentials and pilot-wave forces, we construct the solutions as a dynamics evolving on suitable path spaces. Then, under the assumption that the pilot-wave force is dominated by the potential, we demonstrate that the walker possesses a unique statistical steady state. We conclude by presenting an example of such an invariant measure, as obtained from a numerical simulation of a walker in a harmonic potential.