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 nonlinearities, 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.
