  Microscopic swimming particles, which dissipate energy to execute persistent
directed motion, are a classic example of a non-equilibrium system. We
investigate the non-interacting Ornstein--Uhlenbeck Particle (OUP), which is
propelled through a viscous medium by a force which is correlated over a finite
time. We obtain an exact expression for the steady state phase-space density of
a single OUP confined by a quadratic potential, and use the result to explore
more complex geometries, both through analytical approximations and numerical
simulations. In a "Casimir"-style setup involving two narrowly-spaced walls, we
describe a particle-trapping phenomenon, which leads to a repulsive effective
interaction between the walls; while in a two-dimensional annulus geometry, we
observe net stresses which resemble the Laplace pressure.
