Brownian motion with multiplicative noises revisited

Abstract

The Langevin equation with multiplicative noise and statedependent transport coefficient has to be always complemented with the proper interpretation rule of the noise, such as the Itô and Stratonovich conventions. Although the mathematical relationship between the different rules and how to translate from one rule to another are well-established, it still remains controversial what is a more physically natural rule. In this communication, we derive the overdamped Langevin equation with multiplicative noise for Brownian particles, by systematically eliminating the fast degrees of freedom of the underdamped Langevin equation. The Langevin equations obtained here vary depending on the choice of the noise conventions but they are different representations for an identical phenomenon. The results apply to multi-variable, nonequilibrium, non-stationary systems, and other general settings.

Temporal disorder as a mechanism for spatially heterogeneous diffusion

“…This could arise, for example, in the case of nonlinear Brownian motion where the diffusivity depends on position. The interpretation of multiplicative noise is ambiguous due to the subtleties of stochastic integration, and the associated distinction between Ito and Stratonovich versions of stochastic calculus [3][4][5]. The different interpretations of multiplicative noise result in different versions of the corresponding Fokker-Planck (FP) equation, which describes the evolution of the distribution of sample paths.…”

“…for 0 μ 1. The particular choices μ = 0,1/2,1 correspond, respectively, to the Ito, Stratonovich, and kinetic interpretations [3][4][5]18]. In the case of the piecewise constant diffusivity shown in Fig.…”

“…2is supplemented by an absorbing boundary condition at x = 0 and a reflecting boundary condition at x = L: p(0,t) = 0, J (L,t) = 0. Following the recent analysis of first-passage 2470-0045/2017/95(6)/060101 (5) 060101-1 ©2017 American Physical Society…”

Overdamped limit and inverse-friction expansion for Brownian motion in an inhomogeneous medium

“…And their results have never been tested against numerical simulations. These might cause some confusions, which triggered several recent works on this already resolved Ito-Stratonovich dilemma [4][5][6][7][8][9].For a simpler case with a constant friction coefficient (still space-dependent local temperature), the overdamped limit was rigorously derived by the Fokker-Planck equation approach [10] and also by the Langevin equation approach at the level of a single realization [11]. This case turns out to correspond to the naive Langevin description with the Ito calculus.…”

“…And their results have never been tested against numerical simulations. These might cause some confusions, which triggered several recent works on this already resolved Ito-Stratonovich dilemma [4][5][6][7][8][9].…”