{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathbb{T}$ be $\\Z$, $\\N$, $\\R$ or $[0, \\infty)$, and let $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]].\nLet $(\\theta_t)_{t \\in \\mathbb{T}}$ be a family of [[measure-preserving maps|556WBDAHNP]] $\\theta_t \\colon \\Omega \\to \\Omega$ such that $(t, \\omega) \\mapsto \\theta_t \\omega$ is measurable, $\\theta_0 = \\id$ and $\\theta_{s + t} = \\theta_s \\circ \\theta_t$ for all $s, t \\in \\mathbb{T}$.\nLet $X$ be a measurable space.\nA *random dynamical system* on $X$ over $\\theta$ is a measurable map $\\varphi \\colon \\mathbb{T} \\times \\Omega \\times X \\to X$, written $\\varphi(t, \\omega) = \\varphi(t, \\omega, \\cdot)$, such that for all $\\omega \\in \\Omega$:\n\n1. $\\varphi(0, \\omega) = \\id_X$;\n2. $\\varphi(s + t, \\omega) = \\varphi(t, \\theta_s \\omega) \\circ \\varphi(s, \\omega)$ for all $s, t \\in \\mathbb{T}$.\n\n## Remarks\n\nClause 2 is the cocycle property.\nWhen $\\Omega$ has one point, $\\varphi$ is an ordinary [[dynamical system|BBZP1R872N]].\nOften $X$ is a topological space and each $\\varphi(t, \\omega)$ is required to be continuous.\n\nExample: let $f_0, f_1 \\colon X \\to X$ be measurable maps.\nLet $\\Omega = \\set{0, 1}^{\\Z}$ with the product of the measures giving weight $\\frac{1}{2}$ to $0$ and to $1$, and let $\\theta_t$ be the $t$-th power of the left shift, for $t \\in \\N$.\nPut $\\varphi(0, \\omega) = \\id_X$ and $\\varphi(n, \\omega) = f_{\\omega_{n-1}} \\circ \\cdots \\circ f_{\\omega_1} \\circ f_{\\omega_0}$ for $n \\ge 1$.\nThen $\\varphi(m + n, \\omega)$ applies $f_{\\omega_0}, \\ldots, f_{\\omega_{m-1}}$ and then $f_{\\omega_m}, \\ldots, f_{\\omega_{m+n-1}}$, and the second block is $\\varphi(n, \\theta_m \\omega)$, so clause 2 holds.\n\nNon-example: with the same data, $\\psi(n, \\omega) = f_{\\omega_0}^{\\,n}$ fails clause 2 when $f_0^{\\,2} \\ne f_1 \\circ f_0$.\nIndeed, for $\\omega_0 = 0$ and $\\omega_1 = 1$, $\\psi(2, \\omega) = f_0 \\circ f_0$, while $\\psi(1, \\theta_1 \\omega) \\circ \\psi(1, \\omega) = f_1 \\circ f_0$.\n\nStochastic differential equations with suitable coefficients generate random dynamical systems with $\\Omega$ the space of Brownian paths and $\\theta_t$ the shift of paths.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "DMTFJY6CS0",
    "kind": "definition",
    "msc": [
      "37H05",
      "37H10"
    ],
    "name": "Random dynamical system",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Random_dynamical_system"
      }
    ],
    "tag": "YEFR2GGGJN",
    "uses": [
      "0NFPJVJJHX",
      "556WBDAHNP",
      "BBZP1R872N"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/YEFR2GGGJN/",
  "links": [
    {
      "field": "defines",
      "target": "DMTFJY6CS0"
    },
    {
      "field": "uses",
      "target": "0NFPJVJJHX"
    },
    {
      "field": "uses",
      "target": "556WBDAHNP"
    },
    {
      "field": "uses",
      "target": "BBZP1R872N"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YEFR2GGGJN.md",
  "msc": [
    "37H05",
    "37H10"
  ],
  "name": "Random dynamical system",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YEF/YEFR2GGGJN.md",
  "status": null,
  "tag": "YEFR2GGGJN",
  "uri": "m2://YEFR2GGGJN",
  "url": "https://tamnd.github.io/math2/wiki/random-dynamical-system-definition/",
  "writing": {
    "tag": "YEFR2GGGJN",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}