A method to solve stochastic partial differential equations on random domains consists in using a one-to-one random mapping function which transforms the random domain into a deterministic domain. With this method, the randomness is then borne by the constitutive relationship of the material. In this paper, this method is applied in electrokinetics in the case of scalar potential and vector potential formulations. An example is treated and the proposed method is compared to a nonintrusive method (NIM) based on the remeshing of the random domains.