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  "body": "\n## Definition\n\nLet $(X, \\mathcal{A}, m)$ be a $\\sigma$-finite [[measure|7V9TSDJFZR]] space.\nLet $H = L^2(X, m)$ be the real [[Hilbert space|HM2HHE3J51]] of square-integrable functions (an [[Lp space|SASP14MA1Z]]), with inner product $(\\cdot, \\cdot)$.\nLet $\\mathcal{F}$ be a dense linear subspace of $H$ and let $\\mathcal{E} \\colon \\mathcal{F} \\times \\mathcal{F} \\to \\R$ be a [[bilinear form|0SQZMXB6GP]].\nThe pair $(\\mathcal{E}, \\mathcal{F})$ is a *Dirichlet form* on $L^2(X, m)$ if:\n\n1. $\\mathcal{E}$ is symmetric: $\\mathcal{E}(u, v) = \\mathcal{E}(v, u)$ for all $u, v \\in \\mathcal{F}$.\n2. $\\mathcal{E}$ is nonnegative: $\\mathcal{E}(u, u) \\ge 0$ for all $u \\in \\mathcal{F}$.\n3. $\\mathcal{E}$ is closed: $\\mathcal{F}$ is complete for the norm $\\norm{u}_1 = \\left(\\mathcal{E}(u, u) + (u, u)\\right)^{1/2}$.\n4. $\\mathcal{E}$ is Markovian: for every $u \\in \\mathcal{F}$, the function $v = \\min(\\max(u, 0), 1)$ lies in $\\mathcal{F}$ and $\\mathcal{E}(v, v) \\le \\mathcal{E}(u, u)$.\n\n## Remarks\n\nExample: on $L^2(\\R^n)$ with Lebesgue measure, take $\\mathcal{F}$ to be the [[Sobolev space|M4CEZ5K9HP]] $W^{1,2}(\\R^n)$ and $\\mathcal{E}(u, v) = \\int_{\\R^n} \\nabla u \\cdot \\nabla v \\, dx$.\nIt is symmetric and nonnegative, and $\\norm{\\cdot}_1$ is the Sobolev norm, which is complete.\nThe truncation $v = \\min(\\max(u, 0), 1)$ of $u \\in W^{1,2}$ lies in $W^{1,2}$, with $\\nabla v = \\nabla u$ almost everywhere on $\\set{0 \u003c u \u003c 1}$ and $\\nabla v = 0$ almost everywhere elsewhere, so $\\mathcal{E}(v, v) \\le \\mathcal{E}(u, u)$.\n\nNon-example: on $L^2(\\R)$ take $\\mathcal{F} = W^{2,2}(\\R)$ and $\\mathcal{E}(u, v) = \\int u'' v'' \\, dx$.\nClauses 1 to 3 hold, but clause 4 fails.\nIf $u \\in W^{2,2}(\\R)$ is smooth and crosses the level $1$ with nonzero slope, its truncation has a corner there, so it is not in $W^{2,2}(\\R)$.\n\nA closed, symmetric, nonnegative, densely defined form corresponds to a nonnegative [[self-adjoint operator|0NEGN5CRSC]] $A$ with $\\mathcal{E}(u, v) = (A^{1/2} u, A^{1/2} v)$ and $\\mathcal{F}$ the domain of $A^{1/2}$.\nClause 4 holds if and only if the semigroup $T_t = e^{-tA}$ is Markovian: $0 \\le f \\le 1$ almost everywhere implies $0 \\le T_t f \\le 1$ almost everywhere, for every $t \u003e 0$.\nLet $X$ be a locally compact separable metric space and $m$ a Radon measure of full support.\nThe form is then *regular* if $\\mathcal{F} \\cap C_c(X)$ is dense in $\\mathcal{F}$ for $\\norm{\\cdot}_1$ and dense in $C_c(X)$ for the uniform norm.\n",
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