Approximate Fokker–Planck equation of system driven by multiplicative colored noises with colored cross-correlation

Using the Hanggi ansatz to truncate the evolution equation for probability density, an approximate Fokker–Planck equation (AFPE) is derived. This AFPE is valid for one-dimensional general systems driven by two multiplicative colored noises (τ1≠0 and τ2≠0) that are correlated in color (τ3≠0) under the condition for τ1,τ2, and τ3 to satisfy some inequalities. We apply this AFPE to a symmetrical bistable potential system driven by a colored multiplicative noise and a white additive noise with white cross-correlation. To verify the validity of our analytical approximation the numerical simulations for this system is performed. We discovered a new phenomenon that the symmetry of stationary probability density broken by the correlation between the noises can be gradually recovered as the noise self-correlation time is increased.