Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski-Kramers limit

TL;DR

This paper investigates the limiting behavior of stochastic systems with arbitrary friction and diffusion as inertia vanishes, revealing that the effective dynamics depend on the interpretation of stochastic calculus, including novel position-dependent and unconventional integrals.

Contribution

It introduces a comprehensive analysis of the Smoluchowski-Kramers limit for systems with arbitrary friction and diffusion, highlighting the dependence of the limiting drift on stochastic integral interpretation.

Findings

Different drift fields emerge depending on the stochastic integral interpretation.

Position-dependent and unconventional stochastic integrals are possible in the limit.

Numerical simulations support the theoretical results.

Abstract

We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study the limit where friction effects dominate the inertia, i.e. where the mass goes to zero (Smoluchowski-Kramers limit). {Using the It\^o stochastic integral convention,} we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. {Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation}, which can be parametrized by $\alpha \in \mathbb{R}$. Interestingly, in addition to the classical It\^o ($\alpha=0$), Stratonovich ($\alpha=0.5$) and anti-It\^o ($\alpha=1$) integrals, we show that position-dependent $\alpha = \alpha(x)$, and even stochastic integrals with $\alpha \notin [0,1]$ arise. Our findings are supported by numerical simulations.