{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $E$ be a locally compact Hausdorff space with a countable base, and let $E_\\Delta = E \\cup \\set{\\Delta}$ be its one-point compactification; the point $\\Delta$ is the cemetery.\nLet $X = (X_t)_{t \\ge 0}$ be a [[stochastic process|0QH5ENKZM1]] with values in $E_\\Delta$, adapted to a [[filtration|HNK1Y62G7V]] $(\\mathcal{F}_t)$, with a family of probability measures $(P^x)_{x \\in E_\\Delta}$.\n$X$ is a *Hunt process* if:\n\n1. $X$ is a Markov process with respect to $(\\mathcal{F}_t)$ under each $P^x$, with $P^x(X_0 = x) = 1$, and $\\Delta$ is absorbing: $X_t = \\Delta$ implies $X_s = \\Delta$ for all $s \\ge t$.\n2. Each path $t \\mapsto X_t$ is right-continuous on $[0, \\infty)$ and has left limits in $E_\\Delta$ on $(0, \\infty)$.\n3. $X$ has the strong [[Markov property|0B77KZCQ51]] at every [[stopping time|WND87FBAKC]] of $(\\mathcal{F}_t)$, where $(\\mathcal{F}_t)$ is right-continuous and suitably completed.\n4. $X$ is quasi-left-continuous: if $T_n$ are stopping times increasing to $T$, then $X_{T_n} \\to X_T$ almost surely on $\\set{T \u003c \\infty}$, under each $P^x$.\n\n## Remarks\n\nThe time $\\zeta = \\inf\\set{t : X_t = \\Delta}$ is the lifetime of the process.\nExample: [[Brownian motion|22735FGJV6]] in $\\R^d$ is a Hunt process, with $\\zeta = \\infty$; more generally every [[Feller process|4HS3C12472]] with right-continuous paths having left limits is a Hunt process.\nNon-example: on $E = \\R$, let the process move deterministically at unit speed, except that it jumps up by $1$ when it reaches the point $1$ from below.\nIt is a strong Markov process with right-continuous paths, but clause 4 fails.\nStarted at $0$, the stopping times $T_n = 1 - 1/n$ increase to $T = 1$, while $X_{T_n} \\to 1$ and $X_T = 2$.\nQuasi-left-continuity allows jumps only at totally inaccessible times, such as the jump times of a Poisson process.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "Z7T669BQ9B",
    "kind": "definition",
    "msc": [
      "60J45",
      "60J25",
      "60J40"
    ],
    "name": "Hunt process",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hunt_process"
      }
    ],
    "tag": "HR2GM9DSDJ",
    "uses": [
      "0QH5ENKZM1",
      "0B77KZCQ51",
      "WND87FBAKC",
      "HNK1Y62G7V",
      "22735FGJV6",
      "4HS3C12472"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/HR2GM9DSDJ/",
  "links": [
    {
      "field": "defines",
      "target": "Z7T669BQ9B"
    },
    {
      "field": "uses",
      "target": "0QH5ENKZM1"
    },
    {
      "field": "uses",
      "target": "0B77KZCQ51"
    },
    {
      "field": "uses",
      "target": "WND87FBAKC"
    },
    {
      "field": "uses",
      "target": "HNK1Y62G7V"
    },
    {
      "field": "uses",
      "target": "22735FGJV6"
    },
    {
      "field": "uses",
      "target": "4HS3C12472"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HR2GM9DSDJ.md",
  "msc": [
    "60J45",
    "60J25",
    "60J40"
  ],
  "name": "Hunt process",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HR2/HR2GM9DSDJ.md",
  "status": null,
  "tag": "HR2GM9DSDJ",
  "uri": "m2://HR2GM9DSDJ",
  "url": "https://tamnd.github.io/math2/wiki/hunt-process-definition/",
  "writing": {
    "tag": "HR2GM9DSDJ",
    "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
      }
    ]
  }
}