A quantitative dichotomy for Lyapunov exponents of non-dissipative SDEs with an application to electrodynamics

TL;DR

This paper establishes a clear dichotomy for the top Lyapunov exponent in certain non-dissipative stochastic differential equations, showing it is either always zero or always positive, with implications for charged particle dynamics in magnetic fields.

Contribution

It introduces a quantitative dichotomy for the Lyapunov exponent in non-dissipative SDEs and analyzes its behavior in the small noise limit, with applications to electrodynamics.

Findings

Lyapunov exponent is either zero or positive for all noise levels.

The decay rate of the Lyapunov exponent in the small-noise limit is at most linear.

Application to charged particle motion in fluctuating magnetic fields.

Abstract

In this paper we derive a quantitative dichotomy for the top Lyapunov exponent of a class of non-dissipative SDEs on a compact manifold in the small noise limit. Specifically, we prove that in this class, either the Lyapunov exponent is zero for all noise strengths, or it is positive for all noise strengths and that the decay of the exponent in the small-noise limit cannot be faster than linear in the noise parameter. As an application, we study the top Lyapunov exponent for the motion of a charged particle in randomly-fluctuating magnetic fields, which also involves an interesting geometric control problem.