Langevin动力学的可逆性、协方差与粗粒化：关于乘性噪声的选择
Reversibility, covariance and coarse-graining for Langevin dynamics: On the choice of multiplicative noise
AI总结：
研究SDEs中可逆性、几何与乘性噪声的相互作用，推导了关于Gibbs测度可逆的代数条件，并证明慢快系统在粗粒化后有效动力学保留Klimontovich噪声解释与可逆性。
摘要：
我们研究随机微分方程（SDEs）中可逆性、几何与乘性噪声选择（特别是Itô、Stratonovich、Klimontovich）之间的相互作用。基于统一的几何框架，我们推导了扩散过程在Riemannian流形上关于Gibbs测度可逆的代数条件。该条件连续依赖于参数$λ\in [0,1]$，该参数在Itô（$λ= 0$）、Stratonovich（$λ= \frac 1 2$）和Klimontovich（$λ= 1$）的约定之间进行插值。对于具有块对角扩散结构的可逆慢快SDEs系统，我们利用Dirichlet形式理论证明，可逆性与Klimontovich噪声解释在粗粒化下均得以保持。特别地，我们证明通过投影到低维流形获得的慢变量有效动力学，保留了Klimontovich解释，并且仍关于边缘Gibbs测度/自由能保持可逆。我们的结果为对具有非平凡几何与噪声结构的粗粒化可逆动力学进行建模提供了灵活的变分框架。
英文摘要：
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.
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