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We study the autophoretic motion of a spherical active particle interactingchemically and hydrodynamically with its fluctuating environment in the limitof rapid diffusion and slow viscous flow. Then, the chemical and hydrodynamicfields can be expressed in terms of integrals over boundaries. The resultingboundary-domain integral equations provide a direct way of obtaining thetraction on the particle, requiring the solution of linear integral equations.An exact solution for the chemical and hydrodynamic problems is obtained for aparticle in an unbounded domain. For motion near boundaries, we providecorrections to the unbounded solutions in terms of chemical and hydrodynamicGreen's functions, preserving the dissipative nature of autophoresis in aviscous fluid for all physical configurations. Using this, we give the fullystochastic update equations for the Brownian trajectory of an autophoreticparticle in a complex environment. Finally, we demonstrate our method bystudying autophoresis near a plane interface of fixed viscosity ratio andsolute permeability. We provide explicit solutions to the chemical andhydrodynamic problems with high accuracy for this system geometry. We apply ourtheoretical results in numerical simulations of the dynamics of a bottom-heavyBrownian Janus particle near a wall.

Microhydrodynamics of an autophoretic particle near a plane interface