{
  "id": "2210.10681",
  "version": "v1",
  "published": "2022-10-19T15:52:45.000Z",
  "updated": "2022-10-19T15:52:45.000Z",
  "title": "The Isochronal Phase of Stochastic PDE and Integral Equations: Metastability and Other Properties",
  "authors": [
    "Zachary P. Adams",
    "James MacLaurin"
  ],
  "comment": "40 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We study the dynamics of waves, oscillations, and other spatio-temporal patterns in stochastic evolution systems, including SPDE and stochastic integral equations. Representing a given pattern as a smooth, stable invariant manifold of the deterministic dynamics, we reduce the stochastic dynamics to a finite dimensional SDE on this manifold using the isochronal phase. The isochronal phase is defined by mapping a neighbourbhood of the manifold onto the manifold itself, analogous to the isochronal phase defined for finite-dimensional oscillators by A.T.~Winfree and J.~Guckenheimer. We then determine a probability measure that indicates the average position of the stochastic perturbation of the pattern/wave as it wanders over the manifold. It is proved that this probability measure is accurate on time-scales greater than $O(\\sigma^{-2})$, but less than $O(\\exp(C\\sigma^{-2}))$, where $\\sigma\\ll1$ is the amplitude of the stochastic perturbation. Moreover, using this measure, we determine the expected velocity of the difference between the deterministic and stochastic motion on the manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-19T15:52:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "35R60"
    ],
    "keywords": [
      "isochronal phase",
      "stochastic pde",
      "probability measure",
      "stochastic perturbation",
      "metastability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}