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  "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 $d = \\set{(x, x) : x \\in X}$ be the diagonal of $X \\times X$.\nThen there are\n1. a symmetric form $\\mathcal{E}^{(c)}$ on $\\mathcal{F} \\cap C_c(X)$ with the strong local property: $\\mathcal{E}^{(c)}(u, v) = 0$ whenever $v$ is constant on a neighbourhood of the support of $u$;\n2. a symmetric positive Radon measure $J$ on $X \\times X \\setminus d$, the jumping measure;\n3. a positive Radon measure $k$ on $X$, the killing measure;\n\nsuch that for all $u, v \\in \\mathcal{F} \\cap C_c(X)$\n$$\\mathcal{E}(u, v) = \\mathcal{E}^{(c)}(u, v) + \\iint_{X \\times X \\setminus d} \\big(u(x) - u(y)\\big)\\big(v(x) - v(y)\\big) \\, J(dx, dy) + \\int_X u(x) v(x) \\, k(dx).$$\nThe form $\\mathcal{E}^{(c)}$ and the measures $J$ and $k$ are uniquely determined by $\\mathcal{E}$.\n\n## Remarks\n\nThe proof starts from the Markov property, which gives $\\mathcal{E}(u, v) \\le 0$ for nonnegative $u, v \\in \\mathcal{F} \\cap C_c(X)$ with disjoint supports.\nSo $-\\mathcal{E}(u, v)$ is a positive functional of such pairs, and a Riesz representation argument turns it into the measure $J$.\nThe killing measure comes from the behaviour of $\\mathcal{E}$ on functions equal to $1$ near a compact set, and $\\mathcal{E}^{(c)}$ is what remains.\nSome texts put a factor $2$ or $\\tfrac12$ in front of the jump term, which only rescales $J$.\nFor the form $\\int_{\\R^n} \\nabla u \\cdot \\nabla v \\, dx$ on $W^{1,2}(\\R^n)$, the form is all local: $J = 0$ and $k = 0$.\nFor the fractional form $\\iint (u(x) - u(y))(v(x) - v(y)) \\abs{x - y}^{-n - \\alpha} \\, dx \\, dy$ with $0 \u003c \\alpha \u003c 2$, only the jump part is present.\nFor a Hunt process attached to $\\mathcal{E}$, the three parts describe its continuous motion, its jumps and its killing.\n\n## Explanations\n\n### L1 human\n\nA Dirichlet form measures the energy of a function, and this formula splits that energy into three kinds.\nOne kind comes from smooth local variation, as with the gradient of a function.\nOne kind comes from differences between values at distant points, which correspond to jumps of a random particle.\nThe last kind comes from the function's size alone, which corresponds to the particle being killed.\n\n### L2 human\n\nThe formula is the analytic counterpart of the Lévy–Khintchine decomposition of a Lévy process into a diffusion part, a jump part and a killing rate.\nA regular Dirichlet form is local, meaning that $J = 0$, exactly when its associated Hunt process has continuous paths up to its lifetime.\nIn practice one reads off $J$ and $k$ from a given form to understand the process, or one builds forms from chosen $J$ and $k$ to construct jump processes.\n",
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