Abstract The differential equation governing the convective diffusion of arbitrarily shaped Brownian particles whose mobility is nonisotropic is dervied. The nonisotropic mobility may arise from the presence of a boundary, for example. It is postulated that the thermal fluctuation couples with the nonisotropic mobility to produce an additional force which is biased toward the mobility gradient. The Langevin equation modified to include this nonisotropic force is the starting point our derivation. A generalized Fick's law of diffusion and a generalized Einstein-Stokes relation between the diffusion coefficient and the variable mobility are derived in a six-dimensional linear-angular displacement hyperspace.