  The traction on the surface of a spherical active colloid in a thermally
fluctuating Stokesian fluid contains passive, active, and Brownian
contributions. Here we derive these three parts systematically, by "projecting
out" the fluid using the boundary-domain integral representation of slow
viscous flow. We find an exact relation between the statistics of the Brownian
traction and the thermal forces in the fluid and derive, thereby,
fluctuation-dissipation relations for every irreducible tensorial harmonic
traction mode. The first two modes give the Brownian force and torque, from
which we construct the Langevin and Smoluchowski equations for the position and
orientation of the colloid. We emphasize the activity-induced breakdown of
detailed balance and provide a prescription for computing the
configuration-dependent variances of the Brownian force and torque. We apply
these general results to an active colloid near a plane wall, the simplest
geometry with configuration-dependent variances, and show that the stationary
distribution is non-Gibbsian. We derive a regularization of the translational
and rotational friction tensors, necessary for Brownian dynamics simulations,
that ensures positive variances at all distances from the wall. The many-body
generalization of these results is indicated.
