The isomorphism between electrostatics and diffusion is discussed and utilized to develop a Brownian dynamics algorithm for solving the Poisson equation near dielectric interfaces. The electrostatic potential behaves as if carried by noninteracting, randomly moving pseudo-particles whose residence time in a given region of space is proportional to the electrostatic potential there. By applying random numbers from the exact solution for diffusion near a planar discontinuity, the Brownian motion of these particles can be propagated for large time steps, independent of spatial grids or artificial boundary conditions. The applicability of the Brownian algorithm is demonstrated in simple illustrative calculations.