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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.03347 |
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| _version_ | 1866911250099732480 |
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| author | Ayala, Mario Dirr, Nicolas Pavliotis, Grigorios A. Zimmer, Johannes |
| author_facet | Ayala, Mario Dirr, Nicolas Pavliotis, Grigorios A. Zimmer, Johannes |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03347 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reversibility, covariance and coarse-graining for Langevin dynamics: On the choice of multiplicative noise Ayala, Mario Dirr, Nicolas Pavliotis, Grigorios A. Zimmer, Johannes Probability 60H10 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. |
| title | Reversibility, covariance and coarse-graining for Langevin dynamics: On the choice of multiplicative noise |
| topic | Probability 60H10 |
| url | https://arxiv.org/abs/2511.03347 |