{
  "id": "2104.13181",
  "version": "v1",
  "published": "2021-04-27T13:40:34.000Z",
  "updated": "2021-04-27T13:40:34.000Z",
  "title": "Some asymptotic properties of random walks on homogeneous spaces",
  "authors": [
    "Timothée Bénard"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $G$ be a connected semisimple real Lie group with finite center, and $\\mu$ a probability measure on $G$ whose support generates a Zariski-dense subgroup of $G$. We consider the right $\\mu$-random walk on $G$ and show that each random trajectory spends most of its time at bounded distance of a well-chosen Weyl chamber. We infer that if $G$ has rank one, and $\\mu$ has a finite first moment, then for any discrete subgroup $\\Lambda \\subseteq G$, the $\\mu$-walk and the geodesic flow on $\\Lambda \\backslash G$ are either both transient, or both recurrent and ergodic, thus extending a well known theorem due to Hopf-Tsuji-Sullivan-Kaimanovich dealing with the Brownian motion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-27T13:40:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random walk",
      "asymptotic properties",
      "homogeneous spaces",
      "connected semisimple real lie group",
      "random trajectory spends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}