{
  "id": "2307.07472",
  "version": "v1",
  "published": "2023-07-14T16:54:09.000Z",
  "updated": "2023-07-14T16:54:09.000Z",
  "title": "Spectral gap for projective processes of linear SPDEs",
  "authors": [
    "Martin Hairer",
    "Tommaso Rosati"
  ],
  "comment": "74 pages, 2 figures",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "This work studies the angular component $ \\pi_{t} = u_{t} / \\| u_{t} \\| $ associated to the solution $ u $ of a vector-valued linear hyperviscous SPDE on a $d$-dimensional torus $$\\mathrm{d} u^{\\alpha} =- \\nu^{\\alpha} (- \\Delta)^{\\mathbf{a} } u^{\\alpha} \\mathrm{d} t + (u \\cdot \\mathrm{d} W)^{\\alpha} \\;,\\quad \\alpha \\in \\{ 1, \\dots, m \\} $$ for $ u \\colon \\mathbb{T}^{d} \\to \\mathbb{R}^{m} $, $ \\mathbf{a} \\geqslant 1 $ and a sufficiently smooth and non-degenerate noise $ W $. We provide conditions for existence, as well as uniqueness and spectral gaps (if $ \\mathbf{a} > d/2$) of invariant measures for $ \\pi $ in the projective space. Our proof relies on the introduction of a novel Lyapunov functional for $\\pi_{t}$, based on the study of dynamics of the ``energy median'': the energy level $M$ at which projections of $u$ onto frequencies with energies less or more than $M$ have about equal $L^2$ norm. This technique is applied to obtain -- in an infinite-dimensional setting without order preservation -- lower bounds on top Lyapunov exponents of the equation, and their uniqueness via Furstenberg-Khasminskii formulas.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-14T16:54:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15"
    ],
    "keywords": [
      "spectral gap",
      "linear spdes",
      "projective processes",
      "novel lyapunov functional",
      "furstenberg-khasminskii formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 74,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}