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  "body": "\n## Definition\n\nLet $(E, \\mathcal{E})$ be a measurable space with a σ-finite [[measure|PR11BZQQPG]] $m$.\nLet $(P_t)$ be a sub-Markov transition semigroup on $E$ with [[potential kernel|595AZJAZNA]] $U$.\nA *Green's function* (or potential density) of $(P_t)$ with respect to $m$ is a measurable function $u \\colon E \\times E \\to [0, \\infty]$ such that\n$$U(x, A) = \\int_A u(x, y) \\, m(dy) \\qquad \\text{for every } x \\in E \\text{ and } A \\in \\mathcal{E}.$$\n\n## Remarks\n\nFor each $x$ the function $u(x, \\cdot)$ is determined only $m$-almost everywhere.\nOne usually fixes a version for which $x \\mapsto u(x, y)$ is [[excessive|46436KE7NT]] for each $y$; under standard duality hypotheses such a version exists and is unique.\nExample: for [[Brownian motion|22735FGJV6]] in $\\R^3$ with Lebesgue measure, $u(x, y) = 1/(2\\pi\\abs{x - y})$, because this is the density of $U(x, \\cdot)$.\nFor Brownian motion killed on leaving a planar domain $D$, the Green's function is $u = g_D/\\pi$.\nHere $g_D$ is the classical [[Green's function|M36SD56MWQ]] with $\\Delta g_D(\\cdot, y) = -2\\pi\\delta_y$, and the factor comes from the generator $\\frac12 \\Delta$.\nFor a [[Markov chain|0G48PYM96Q]] on a countable set with counting measure, $u(x, y) = \\sum_{n \\ge 0} P^n(x, y)$ is the expected number of visits to $y$ starting from $x$.\nNon-example: Brownian motion in $\\R^2$ on the whole plane has no Green's function with finite values, since $U(x, A) = \\infty$ for every nonempty open $A$.\n",
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