Stochastic Fokker-Planck equation in random environments

We analyze the stochastic dynamics of a large population of noninteracting particles driven by a common environmental input in the form of an Ornstein-Uhlenbeck (OU) process. The density of particles evolves according to a stochastic Fokker-Planck (FP) equation with respect to different realizations of the OU process. We then exploit the connection with previous work on diffusion in randomly switching environments in order to derive moment equations for the distribution of solutions to the stochastic FP equation. We use perturbation theory and Green's functions to calculate the mean and variance of the distribution when the relaxation rate of the OU process is fast (close to the white-noise limit). Finally, we show how the theory of noise-induced synchronization can be recast into the framework of a stochastic FP equation.

Figure 1

Diagram illustrating the difference between the particle and population perspectives. (a) Multiple realizations of a single particle moving in a random environment generates the probability density $p(x,\alpha ,t)$. (b) Large population $\left(N\to \infty \right)$ of particles evolving in a single realization $\sigma$ of the common random environment generates the population density $P(x,t)$. (c) The stochastic FP equation describes the evolution of the population density $P(x,t)$ for a given realization $\sigma$ of the noisy input $\alpha \left(t\right)$.