{
  "id": "1510.02722",
  "version": "v1",
  "published": "2015-10-09T16:08:56.000Z",
  "updated": "2015-10-09T16:08:56.000Z",
  "title": "Random Walks on Homogeneous Spaces by Sparse Solvable Measures",
  "authors": [
    "C. Davis Buenger"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The paper analyzes a specific class of random walks on quotients of $X:=\\text{SL}(k,{\\Bbb R})/ \\Gamma$ for a lattice $\\Gamma$. Consider a one parameter diagonal subgroup, $\\{g_t\\}$, with an associated abelian expanding horosphere, $U\\cong {\\Bbb R}^k$, and let $\\phi:[0,1]\\rightarrow U$ be a sufficiently smooth curve satisfying the condition that that the derivative of $\\phi$ spends $0$ time in any one subspace of ${\\Bbb R}^k$. Let $ \\mu_U$ be the measure defined as $\\phi_*\\lambda_{[0,1]},$ where $\\lambda_{[0,1]}$ is the Lebesgue measure on $[0,1]$. Let $\\mu_A$ be a measure on the full diagonal subgroup of $\\text{SL}(k,{\\Bbb R})$, such that almost surely the random walk on the diagonal subgroup $A$ with respect to this measure grows exponentially in the direction of the cone expanding $U$. Then the random walk starting at any point $z\\in X$, and alternating steps given by $\\mu_U$ and $\\mu_A$ equidistributes respect to $\\text{SL}(k,{\\Bbb R})$-invariant measure on $X$. Furthermore, the measure defined by $\\mu_A*\\mu_U*\\dots*\\mu_A* \\mu_U*\\delta_z$ converges exponentially fast to the $\\text{SL}(k,{\\Bbb R})$-invariant measure on $X$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-09T16:08:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random walk",
      "sparse solvable measures",
      "homogeneous spaces",
      "invariant measure",
      "full diagonal subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}