{
  "id": "2402.04217",
  "version": "v1",
  "published": "2024-02-06T18:20:47.000Z",
  "updated": "2024-02-06T18:20:47.000Z",
  "title": "Geometric theory of (extended) time-reversal symmetries in stochastic processes -- Part I: finite dimension",
  "authors": [
    "Jérémy O'Byrne",
    "Michael E. Cates"
  ],
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.soft"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-06T18:20:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic process",
      "time-reversal symmetries",
      "geometric theory",
      "finite dimension",
      "reversibility conditions play"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}