Ornstein-Uhlenbeck approximation of one-step processes: a differential equation approach

The steady state of the Fokker-Planck equation corresponding to a density dependent one-step process is approximated by a suitable normal distribution. Starting from the master equations of the process, written in terms of the time dependent probabilities, $p_k(t)$ of the states $k=0,1,\ldots , N$, their continuous (in space) version, the Fokker-Planck equation is formulated. This PDE approximation enables us to create analytic approximation formulas for the steady state distribution. These formulas are derived based on heuristic reasoning and then their accuracy is proved to be of order $1/N^{\beta}$ with some power $\beta <1$.