Colored stochastic multiplicative processes with additive noise unveil a third-order partial differential equation, defying conventional Fokker-Planck equation and Fick-law paradigms

Research on stochastic differential equations (SDEs) involving both additive and multiplicative noise has been extensive. In cases where the primary process is driven by a multiplicative stochastic process, additive white noise typically represents an intrinsic and unavoidable fast component. This applies to phenomena such as thermal fluctuations, inherent uncertainties in measurement processes, or rapid wind forcing in ocean dynamics. In this paper, we focus on an important class of such systems, particularly those characterized by linear drift and multiplicative noise, which have been extensively explored in the literature. In many existing studies, multiplicative stochastic processes are often treated as white noise. However, when considering colored multiplicative noise, the emphasis has usually been on characterizing the far tails of the probability density function (PDF), irrespective of the spectral properties of the noise. In the absence of additive noise and with a general colored multiplicative SDE, standard perturbation approaches lead to a second-order partial differential equation (PDE) known as the Fokker-Planck equation (FPE), consistent with Fick's law. In this paper, we reveal a significant deviation from this standard behavior when additive white noise is introduced. At the leading order of the stochastic process strength, perturbation approaches yield a third-order PDE, regardless of the white noise intensity. The breakdown of the FPE further indicates the breakdown of Fick's law. Additionally, we derive the explicit solution for the equilibrium PDF corresponding to this third-order PDE master equation. Through numerical simulations, we demonstrate significant deviations from results obtained using the FPE derived from Fick's law.