We propose a new model of 2D free particle diffusion on a possibly curved surface. This model is a generalization of the standard Ornstein-Uhlenbeck process and is completely determined by writing down the transport equation describing the diffusion in the phase-space of the diffusing particle. This transport equation is then used to show that curvature effects can profoundly affect the phenomenology of diffusion in the hydrodynamic limit. A specific pedagogical example is also worked out.