  We study the diffusion of particles confined close to a single wall and in
double-wall planar channel geometries where the local diffusivities depend on
the distance to the boundaries. Displacement parallel to the walls is Brownian
as characterized by its variance, but it is non-Gaussian having a non-zero
fourth cumulant. Establishing a link with Taylor dispersion, we calculate the
fourth cumulant and the tails of the displacement distribution for general
diffusivity tensors along with potentials generated by either the walls or
externally, for instance gravity. Experimental and numerical studies of the
motion of a colloid in the direction parallel to the wall give measured fourth
cumulants which are correctly predicted by our theory. Interestingly, contrary
to models of Brownian-yet-non-Gaussian diffusion, the tails of the displacement
distribution are shown to be Gaussian rather than exponential. All together,
our results provide additional tests and constraints for the inference of force
maps and local transport properties near surfaces.
