Statistical properties of a discrete version of the Ornstein-Uhlenbeck process

A discrete version of the Ornstein-Uhlenbeck process is discussed which arises from a simple generalization of the master equation of the random walk. The calculation of the statistical properties of the free propagator for this process can be obtained using essentially the same formalism as for simple random walks. These calculations are carried out in some detail for the one-dimensional case. The usual equation for the evolution of the probability distribution of the Ornstein-Uhlenbeck process is recovered in the continuum limit if the jump distribution has a finite variance. However, the discrete process is also well defined for long tailed jump distributions and, thus, can be used to describe a Lèvy walk under the effect of a harmonic potential. Finally, a brief discussion of the generalization of this process to describe random walks in general potentials is presented and briefly compared with results arising from the fractional diffusion approach.