  Miniaturized, self-propelled locomotors use chemo-mechanical transduction
mechanisms to convert fuel in the environment to autonomous motion. Recent
experimental and theoretical studies demonstrate that these autonomous engines
can passively follow the contours of solid boundaries they encounter. Boundary
guidance, however, is not necessarily stable: Mechanical disturbances can cause
the motor to hydrodynamically depart from the passively guided pathway.
Furthermore, given the scaled-down size of micromotors (typically 100 nm -10
$\mu$m), Brownian thermal fluctuation forces are necessarily important and
these stochastic forces can randomize passively-steered trajectories.
  Here we examine theoretically the stability of boundary guided motion of
micromotors along infinite planar walls to mechanical disturbances and to
Brownian forces. Our aim is to understand under what conditions this passively
guided motion is stable. We choose a locomotor design in which spherical
colloids are partially coated with a catalytic cap that reacts with solute to
produce a product. The product is repelled from the particle surface, causing
the particle to move with the inert face at the front (autonomous motion via
self-diffusiophoresis). When propelled towards a planar wall, deterministic
hydrodynamic studies demonstrate that these locomotors can exhibit, for large
enough cap sizes, steady trajectories in which the particle either skims
unidirectionally along the surface at a constant distance from the wall, or
becomes stationary. We first investigate the linear hydrodynamic stability of
these states by expanding the equations of motion about the states, and find
that linear perturbations decay exponentially in time. We then study the
effects of thermal fluctuations by formulating a Langevin equation for the
particle motion which includes the Brownian stochastic force...
