Article Dans Une Revue Journal of Statistical Mechanics: Theory and Experiment Année : 2024

Geometric theory of (extended) time-reversal symmetries in stochastic processes: I. Finite dimension

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Abstract In this article, we analyze three classes of time-reversal of a Markov process with Gaussian noise on a manifold. We first unveil a commutativity constraint for the most general of these time-reversals to be well defined. Then we give a triad of necessary and sufficient conditions for the stochastic process to be time-reversible. While most reversibility conditions in the literature require knowledge of the stationary probability, our conditions do not, and therefore can be analytically checked in a systematic way. We then show that the mathematical objects whose cancellation is required by our reversibility conditions play the role of independent sources of entropy production. Furthermore, we give a geometric interpretation of the so-called irreversible cycle-affinity as the vorticity of a certain vector field for a Riemannian geometry given by the diffusion tensor. We also discuss the relation between the time-reversability of the stochastic process and that of an associated deterministic dynamics: its Stratonovitch average. Finally, we show that a suitable choice of a reference measure—that can be considered as a prior or a gauge, depending on the context—allows to study a stochastic process in a way that is both coordinate-free and independent of the prescription used to define stochastic integrals. When this reference measure plays the role of a gauge choice, we interpret our previous results through the lens of gauge theory and prove them to be gauge-invariant.
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hal-04930649 , version 1 (05-02-2025)

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J O’byrne, M E Cates. Geometric theory of (extended) time-reversal symmetries in stochastic processes: I. Finite dimension. Journal of Statistical Mechanics: Theory and Experiment, 2024, 2024 (11), pp.113207. ⟨10.1088/1742-5468/ad8f2b⟩. ⟨hal-04930649⟩
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