Learning Equilibrium Fluctuation Expansions from Overdamped Langevin Dynamics

TL;DR

This paper develops higher-order fluctuation expansions for overdamped Langevin dynamics in a double-well potential, analyzing their long-time behavior and recursive structure, with implications for understanding equilibrium limits.

Contribution

It introduces a recursive formula for fluctuation expansion coefficients and studies their long-time limits, extending analysis to multi-dimensional cases.

Findings

In scalar case, coefficients converge exponentially to finite limits.

In higher dimensions, coefficients reflect degeneracy, preventing finite limits.

Recursive formula links dynamical and equilibrium expansion coefficients.

Abstract

We study higher-order small-noise fluctuation expansions for the overdamped Langevin dynamics in a quartic double-well potential. Assuming that the initial data admits a suitable expansion structure, we obtain a strong dynamical expansion of the trajectories, as well as an expansion of the laws with respect to smooth observables. We then investigate the long-time behavior of the expansion coefficients. In the scalar case $d=1$, each coefficient converges exponentially fast to a finite limit as $t\to\infty$. In contrast, for $d\ge 2$, the fluctuation expansion coefficients reflect the degeneracy of the manifold of minima, which in general prevents the existence of a finite long-time limit. Furthermore, by combining a multi-level induction with combinatorial arguments, we derive a recursive formula for the fluctuation expansion coefficients. This recursion shows that the long-time limits of these dynamical expansion coefficients coincide with those arising from the corresponding equilibrium expansions.