Reversibility, covariance and coarse-graining for Langevin dynamics: On the choice of multiplicative noise

TL;DR

This paper explores how different interpretations of multiplicative noise affect reversibility and coarse-graining in Langevin dynamics, providing a geometric framework to understand and preserve these properties in reduced models.

Contribution

It introduces a unified geometric approach to analyze reversibility under various noise interpretations and demonstrates the preservation of reversibility and noise interpretation during coarse-graining.

Findings

Reversibility depends continuously on the noise interpretation parameter.

Coarse-graining preserves Klimontovich interpretation and reversibility in slow-fast systems.

The framework offers a variational approach for modeling reduced reversible dynamics.

Abstract

We study the interplay between reversibility, geometry, and the choice of multiplicative noise (in particular It\^{o}, Stratonovich, Klimontovich) in stochastic differential equations (SDEs). Building on a unified geometric framework, we derive algebraic conditions under which a diffusion process is reversible with respect to a Gibbs measure on a Riemannian manifold. The condition depends continuously on a parameter $\lambda \in [0,1]$ which interpolates between the conventions of It\^o ($\lambda = 0$), Stratonovich ($\lambda = \frac 1 2$) and Klimontovich ($\lambda = 1$). For reversible slow-fast systems of SDEs with a block-diagonal diffusion structure, we show, using the theory of Dirichlet forms, that both reversibility and the Klimontovich noise interpretation are preserved under coarse-graining. In particular, we prove that the effective dynamics for the slow variables, obtained via projection onto a lower-dimensional manifold, retain the Klimontovich interpretation and remain reversible with respect to the marginal Gibbs measure/free energy. Our results provide a flexible variational framework for modeling coarse-grained reversible dynamics with nontrivial geometric and noise structures.