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

TL;DR

We study reversibility of Langevin diffusions with multiplicative noise and geometric structure, focusing on how the stochastic calculus convention ($\lambda$) interacts with a Gibbs measure on a Riemannian manifold. We derive algebraic conditions $(2\lambda-1)\\nabla^c\\cdot(\\sigma\\sigma^T) = 2\lambda \,\\sigma(\\nabla^c\\cdot\\sigma^T)$ for reversibility, specifying It\^{o} ($\\lambda=0$), Stratonovich ($\\lambda=\tfrac{1}{2}$), and Klimontovich ($\\lambda=1$). We prove Klimontovich reversibility and the associated noise interpretation are preserved under coarse-graining via Mosco convergence of Dirichlet forms, enabling a variational framework for coarse-grained reversible dynamics. The results provide guidelines for constructing geometry-aware reversible diffusion models and principled multiscale coarse-graining in high dimensions with multiplicative noise, with potential applications to sampling and multiscale modeling.

Abstract

We study the interplay between reversibility, geometry, and the choice of multiplicative noise (in particular Itô, 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 $λ\in [0,1]$ which interpolates between the conventions of Itô ($λ= 0$), Stratonovich ($λ= \frac 1 2$) and Klimontovich ($λ= 1$). 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.