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

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∈ [0,1] which interpolates between the conventions of Itô ($λ= 0$), Stratonovich (λ= 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.