  We study the autophoretic motion of a spherical active particle interacting
chemically and hydrodynamically with its fluctuating environment in the limit
of rapid diffusion and slow viscous flow. Then, the chemical and hydrodynamic
fields can be expressed in terms of integrals over boundaries. The resulting
boundary-domain integral equations provide a direct way of obtaining the
traction on the particle, requiring the solution of linear integral equations.
An exact solution for the chemical and hydrodynamic problems is obtained for a
particle in an unbounded domain. For motion near boundaries, we provide
corrections to the unbounded solutions in terms of chemical and hydrodynamic
Green's functions, preserving the dissipative nature of autophoresis in a
viscous fluid for all physical configurations. Using this, we give the fully
stochastic update equations for the Brownian trajectory of an autophoretic
particle in a complex environment. Finally, we demonstrate our method by
studying autophoresis near a plane interface of fixed viscosity ratio and
solute permeability. We provide explicit solutions to the chemical and
hydrodynamic problems with high accuracy for this system geometry. We apply our
theoretical results in numerical simulations of the dynamics of a bottom-heavy
Brownian Janus particle near a wall.
