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              <p>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. This system is a well-controlled physical realization of Brownian-yet-non-Gaussian diffusion generated by a fluctuating diffusivity mechanism for which the local diffusivity is quantified. These results provide additional tests and constraints for the inference of  force maps and local transport properties near surfaces.</p>
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