  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.
