Effective drifts in dynamical systems with multiplicative noise: a review of recent progress

Abstract

Noisy dynamical models are employed to describe a wide range of phenomena. Since exact modeling of these phenomena requires access to their microscopic dynamics, whose time scales are typically much shorter than the observable time scales, there is often need to resort to effective mathematical models such as stochastic differential equations (SDEs). In particular, here we consider effective SDEs describing the behavior of systems in the limits when natural time scales become very small. In the presence of multiplicative noise (i.e. noise whose intensity depends upon the system's state), an additional drift term, called noise-induced drift or effective drift, appears. The nature of this noise-induced drift has been recently the subject of a growing number of theoretical and experimental studies. Here, we provide an extensive review of the state of the art in this field. After an introduction, we discuss a minimal model of how multiplicative noise affects the evolution of a system. Next, we consider several case studies with a focus on recent experiments: the Brownian motion of a microscopic particle in thermal equilibrium with a heat bath in the presence of a diffusion gradient; the limiting behavior of a system driven by a colored noise modulated by a multiplicative feedback; and the behavior of an autonomous agent subject to sensorial delay in a noisy environment. This allows us to present the experimental results, as well as mathematical methods and numerical techniques, that can be employed to study a wide range of systems. At the end we give an application-oriented overview of future projects involving noise-induced drifts, including both theory and experiment.

Quantifying non-ergodicity of anomalous diffusion with higher order moments

“…Note that rigorous results on the general symmetries of overdamped multiplicative white noise processes, clarifying some prevailing misconceptions on the Itô-Stratonovich dilemma and presenting an exact mapping of the different interpretations to a purely additive white noise process, were reported only recently 97 . Nevertheless, the notion of a ‘correct’ interpretation of multiplicative white noise Langevin equations is always context dependent 98–100 , as it was shown, for example, that general non-linear relaxation processes not satisfying a fluctuation-dissipation theorem in fact obey a Fokker-Planck equation following from the Itô interpretation of the underlying Langevin equation 101 . Moreover, as we here advocate the general importance of higher-order statistics in the quantification of stationary and/or ergodic properties of single-particle time series data, the specific interpretation of the multiplicative white noise Langevin equation is not critical.…”

Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity

“…Smyshlyaev and Chen applied a position-dependent diffusion coefficient to the boundary feedback control of a diffusive system and proved the Mittag-Leffler stability of the system [45,46]. Several other works have since focussed on position-dependent diffusive systems [47][48][49][50], to name but a few.Differing from a constant diffusion coefficient, the presence of a position-dependent diffusivity involves the problem of how to interpret multiplicative noise in a stochastic equation, particularly, a noise-induced drift, which varies by choosing different integral forms, such as Itô, Stratonovich and isothermal integrals [44,51]. It is found that the probability distribution generated within the isothermal integral formulation effects the required Boltzmann distribution, which is correct in thermal equilibrium state [48][49][50], while the Itô and Stratonovich integrals lead to 'athermal' forms.…”

“…It is found that the probability distribution generated within the isothermal integral formulation effects the required Boltzmann distribution, which is correct in thermal equilibrium state [48][49][50], while the Itô and Stratonovich integrals lead to 'athermal' forms. Generally, the Itô interpretation is employed in economics and biology due to their features of being 'only related to the latest past'; the Stratonovich integral finds applications in physical systems, such as electrical circuits driven by multiplicative noises (see [51], and references therein). In particular, it finds that with decreasing the mass of particles a second-order Langevin equation can reduce to a first-order one with different noise-induced drifts, depending on the relationship between friction and diffusion coefficients [51][52][53].…”

“…Generally, the Itô interpretation is employed in economics and biology due to their features of being 'only related to the latest past'; the Stratonovich integral finds applications in physical systems, such as electrical circuits driven by multiplicative noises (see [51], and references therein). In particular, it finds that with decreasing the mass of particles a second-order Langevin equation can reduce to a first-order one with different noise-induced drifts, depending on the relationship between friction and diffusion coefficients [51][52][53]. For example, a second-order Langevin system with a constant friction coefficient but a position-dependent diffusivity will converge to a first-order equation without noise-induced drift.…”

Master equations and the theory of stochastic path integrals

“…For a discussion of stochastic differential equations the reader may refer to a recent report on progress [112].…”