{
  "id": "2108.09361",
  "version": "v1",
  "published": "2021-08-20T20:43:55.000Z",
  "updated": "2021-08-20T20:43:55.000Z",
  "title": "Random Tessellations and Gibbsian solutions of Hamilton-Jacobi Equations",
  "authors": [
    "Mehdi Ouaki",
    "Fraydoun Rezakhanlou"
  ],
  "comment": "61 pages, 1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We pursue two goals in this article. As our first goal, we construct a family $\\mathcal{M}_G$ of Gibbs like measures on the set of piecewise linear convex functions $g:\\mathbb{R}^2\\to\\mathbb{R}$. It turns out that there is a one-to-one correspondence between the gradient of such convex functions and $\\textit{Laguerre tessellations}$. Each cell in a Laguerre tessellation is a convex polygon that is marked by a vector $\\rho\\in\\mathbb{R}^2$. Each measure $\\nu^f\\in\\mathcal{M}_G$ in our family is uniquely characterized by a kernel $f(x,\\rho^-,\\rho^+)$, which represents the rate at which a line separating two cells associated with marks $\\rho^-$ and $\\rho^+$ passes through $x$. To construct our measures, we give a precise recipe for the law of the restriction of our tessellation to a box. This recipe involves a boundary condition, and a dynamical description of our random tessellation inside the box. As we enlarge the box, the consistency of these random tessellations requires that the kernel satisfies a suitable kinetic like PDE. As our second goal, we study the invariance of the set $\\mathcal{M}_G$ with respect to the dynamics of such Hamilton-Jacobi PDEs. In particular we $\\textit{conjecture}$ the invariance of a suitable subfamily $\\widehat{\\mathcal{M}_G}$ of $\\mathcal{M}_G$. More precisely, we expect that if the initial slope $u_x(\\cdot,0)$ is selected according to a measure $\\nu^{f}\\in \\widehat{\\mathcal{M}_G}$, then at a later time the law of $u_x(\\cdot, t)$ is given by a measure $\\nu^{\\Theta_t(f)}\\in\\widehat{\\mathcal{M}_G}$, for a suitable kernel $\\Theta_t(f)$. As we vary $t$, the kernel $\\Theta_t(f)$ must satisfy a suitable kinetic equation. We remark that the function $u$ is also piecewise linear convex function in $(x,t)$, and its law is an example of a Gibbs-like measure on the set of Laguerre tessellations of certain convex subsets of $\\mathbb{R}^3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-20T20:43:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J76",
      "35F21"
    ],
    "keywords": [
      "hamilton-jacobi equations",
      "gibbsian solutions",
      "piecewise linear convex function",
      "laguerre tessellation",
      "random tessellation inside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}