  We consider a simple model for the fluctuating hydrodynamics of a flexible
polymer in dilute solution, demonstrating geometric ergodicity for a pair of
particles that interact with each other through a nonlinear spring potential
while being advected by a stochastic Stokes fluid velocity field. This is a
generalization of previous models which have used linear spring forces as well
as white-in-time fluid velocity fields.
  We follow previous work combining control theoretic arguments, Lyapunov
functions, and hypo-elliptic diffusion theory to prove exponential convergence
via a Harris chain argument. In addition we allow the possibility of excluding
certain "bad" sets in phase space in which the assumptions are violated but
from which the system leaves with a controllable probability. This allows for
the treatment of singular drifts, such as those derived from the Lennard-Jones
potential, which is a novel feature of this work.
