{
  "id": "2211.08242",
  "version": "v1",
  "published": "2022-11-15T15:56:09.000Z",
  "updated": "2022-11-15T15:56:09.000Z",
  "title": "Exponential ergodicity of stochastic heat equations with Hölder coefficients",
  "authors": [
    "Yi Han"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider ergodicity of stochastic heat equations driven by space-time white noise in dimension one, whose drift and diffusion coefficients are merely H\\\"older continuous. We give a short proof that there exists a unique in law mild solution when the diffusion coefficient is $\\beta$ - H\\\"older continuous for $\\beta>\\frac{3}{4}$ and uniformly nondegenerate, and that the drift is locally H\\\"older continuous. Due to non-smoothness of the coefficients, it is hard to obtain Bismut-Elworthy-Li type estimates that are central to the strong Feller property and unique ergodicity in the Lipschitz case, and transforming the coordinates via solving a Kolmogorov equation also seems intractable. Instead, we construct a distance function $d$ which is locally contracting and that bounded subsets of the state space are small in the sense of Harris' ergodic theorem. The construction utilizes a generalized coupling technique introduced by Kulik and Scheutzow in Ann.Probab vol.48 No.6: 3041-3076, 2020 in the setting of stochastic delay equations. Assuming the existence of a suitable Lyapunov function, we prove existence of a spectral gap and that transition probabilities converge exponentially fast to the unique invariant measure. The same method can be applied when the SPDE has a Burgers type nonlinearity $(-A)^{1/2}F(X_t)$ or $\\partial_x F(X_t)$, where $F$ is continuous and has linear growth. We prove existence of a unique in law mild solution when $F$ is locally $\\zeta$-H\\\"oloder continuous for $\\zeta>\\frac{1}{2}$, and the other assumptions are the same as before. We then extend spectral gap and exponential ergodicity results to this case once we have a suitable Lyapunov function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-15T15:56:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exponential ergodicity",
      "hölder coefficients",
      "law mild solution",
      "suitable lyapunov function",
      "diffusion coefficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}