An overdamped univariate system subjected to spatiotemporal noise in the frame of Fokker–Planck equation

For previous research on stochastic dynamics, investigators focused mainly on systems subjected to temporal noise, and established the Fokker–Planck equation (FPE) corresponding to the Langevin equation describing them. In fact, numerous systems driven by spatiotemporal noise exist across biology, physics, chemistry, and related fields. This study analytically investigates an overdamped univariate spatiotemporal system subjected to spatiotemporal noise in the frame of FPE. First, we employ the Taylor expansion of the spatiotemporal \(\delta\) function to derive the Kramers–Moyal equation. Subsequently, based on the Langevin equation which describes dynamical behavior of the system, the nth-order transition moment of state variable of the system is calculated to acquire the corresponding FPE for the overdamped univariate spatiotemporal stochastic system. Finally, using monostable and bistable systems subjected to spatiotemporal noise as illustrative cases, we demonstrate that their steady state probability distribution functions obtained from the spatiotemporal FPE agree with numerical solutions of the corresponding spatiotemporal Langevin equations, thereby validating the spatiotemporal FPE’s correctness. This research result not only provides an important tool for investigating analytically the systems subjected to the spatiotemporal noise, but also improves further the current stochastic dynamical theories.