{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ be a [[locally compact|0FE2SB8WVJ]] separable metric space and let $m$ be a positive [[Radon measure|HZNEQQBCW4]] on $X$ with full support.\nLet $(\\mathcal{E}, \\mathcal{F})$ be a regular [[Dirichlet form|MH20Q2N1AH]] on $L^2(X, m)$, and let $(T_t)_{t \\ge 0}$ be the strongly continuous symmetric Markovian [[contraction semigroup|XPWE94BV6N]] on $L^2(X, m)$ associated with it.\nThen there is an $m$-symmetric [[Hunt process|Z7T669BQ9B]] $M$ on $X$ whose transition function $(p_t)_{t \\ge 0}$ satisfies, for every $t \u003e 0$ and every bounded Borel function $f \\in L^2(X, m)$:\n$$p_t f = T_t f \\quad m\\text{-almost everywhere}.$$\nSuch a process is unique up to equivalence.\nThat is, two $m$-symmetric Hunt processes with this property have the same transition functions outside a set of zero capacity for $\\mathcal{E}$.\nThis exceptional set can be chosen so that neither process enters it.\n\n## Remarks\n\nThe process is built from the semigroup, but the semigroup only acts on classes of functions defined almost everywhere.\nThe difficulty is to choose good pointwise versions, and this is done with the capacity of $\\mathcal{E}$.\nEvery element of $\\mathcal{F}$ has a quasi-continuous version, which is continuous outside open sets of small capacity.\nRegularity of the form provides enough continuous functions in $\\mathcal{F}$ to separate points and to build the process on $X$ itself.\nMoreover $p_t f$ is a quasi-continuous version of $T_t f$.\nExample: the form $\\tfrac12 \\int_{\\R^n} \\nabla u \\cdot \\nabla v \\, dx$ on $W^{1,2}(\\R^n)$ corresponds to Brownian motion on $\\R^n$.\nMa and Röckner extended the correspondence to quasi-regular Dirichlet forms, which include forms on infinite-dimensional spaces.\n\n## Explanations\n\n### L1 human\n\nA Dirichlet form is an energy for functions, and a Hunt process is a random motion that may jump or be killed.\nThe theorem says that every regular energy of this kind comes from a random motion.\nHeat flow defined by the energy is the same as averaging over the random paths of the process.\nSo questions about the energy can be answered with random paths, and conversely.\n\n### L2 human\n\nThe theorem is a central result of the monograph of Fukushima, Oshima and Takeda on Dirichlet forms.\nTogether with the [[Beurling–Deny formula|EK58S2872F]], it gives a meaning to the three parts of the form.\nThe local part, the jumping measure and the killing measure describe the continuous motion, the jumps and the killing of the process.\nA regular Dirichlet form has the local property exactly when the process is a diffusion, that is, has continuous paths up to its lifetime.\nA nearly Borel set has zero capacity if and only if the process, started from $m$-almost every point, never hits it, which gives a probabilistic meaning to capacity.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "MH20Q2N1AH",
      "Z7T669BQ9B"
    ],
    "kind": "statement",
    "msc": [
      "60J46",
      "31C25",
      "60J25"
    ],
    "name": "Fukushima correspondence between Dirichlet forms and processes",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dirichlet_form"
      }
    ],
    "tag": "5PJF1NDCTT",
    "uses": [
      "MH20Q2N1AH",
      "Z7T669BQ9B",
      "HZNEQQBCW4",
      "0FE2SB8WVJ",
      "XPWE94BV6N",
      "EK58S2872F"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/5PJF1NDCTT/",
  "links": [
    {
      "field": "about",
      "target": "MH20Q2N1AH"
    },
    {
      "field": "about",
      "target": "Z7T669BQ9B"
    },
    {
      "field": "uses",
      "target": "MH20Q2N1AH"
    },
    {
      "field": "uses",
      "target": "Z7T669BQ9B"
    },
    {
      "field": "uses",
      "target": "HZNEQQBCW4"
    },
    {
      "field": "uses",
      "target": "0FE2SB8WVJ"
    },
    {
      "field": "uses",
      "target": "XPWE94BV6N"
    },
    {
      "field": "uses",
      "target": "EK58S2872F"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5PJF1NDCTT.md",
  "msc": [
    "60J46",
    "31C25",
    "60J25"
  ],
  "name": "Fukushima correspondence between Dirichlet forms and processes",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5PJ/5PJF1NDCTT.md",
  "status": null,
  "tag": "5PJF1NDCTT",
  "uri": "m2://5PJF1NDCTT",
  "url": "https://tamnd.github.io/math2/wiki/fukushima-correspondence-between-dirichlet-forms-and-processes/",
  "writing": {
    "tag": "5PJF1NDCTT",
    "kind": "statement",
    "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
      }
    ]
  }
}