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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be a domain with a Green's function, and fix $x_0 \\in \\Omega$.\nLet $M$ be the [[Martin kernel|1DBC9FYF7M]] of $\\Omega$ with reference point $x_0$.\nCall a sequence $(y_k)$ in $\\Omega$ *fundamental* if it eventually leaves each compact subset of $\\Omega$ and the functions $M(\\cdot, y_k)$ converge locally uniformly on $\\Omega$.\nThe *Martin boundary* $\\Delta$ of $\\Omega$ is the set of all limits\n$$h = \\lim_{k \\to \\infty} M(\\cdot, y_k)$$\nof fundamental sequences, and for $\\xi = h \\in \\Delta$ one writes $M(\\cdot, \\xi) = h$.\nThe set $\\Omega \\cup \\Delta$ is the *Martin compactification* of $\\Omega$.\n\n## Remarks\n\nEach $M(\\cdot, \\xi)$ with $\\xi \\in \\Delta$ is a positive [[harmonic function|0DEB0JMN0W]] on $\\Omega$ with value $1$ at $x_0$.\nThe set $\\Omega \\cup \\Delta$ carries a compact metrisable topology that induces the usual topology on $\\Omega$.\nIn it, $\\Omega$ is open and dense, and each fundamental sequence converges to its limit in $\\Delta$.\nSo it is a [[compactification|Q6XNC8JAZD]] of $\\Omega$, and for each fixed $x \\in \\Omega$ the function $y \\mapsto M(x, y)$ extends continuously to $\\Delta$ by $\\xi \\mapsto M(x, \\xi)$.\nIt does not depend on the choice of $x_0$, up to a homeomorphism that is the identity on $\\Omega$.\nA point $\\xi \\in \\Delta$ is *minimal* if $M(\\cdot, \\xi)$ is a [[minimal harmonic function|NA4BTZ6WRB]]; the minimal points form the *minimal Martin boundary* $\\Delta_1$.\nExample: let $B$ be the unit ball of $\\R^n$ and $x_0 = 0$.\nA sequence is fundamental if and only if it converges to a point $\\zeta$ of the sphere, and the limit is the [[Poisson kernel|TCYSWCTYYM]] $P(\\cdot, \\zeta)$.\nDistinct $\\zeta$ give distinct kernels, so $\\Delta$ is homeomorphic to the unit sphere.\nBy contrast, consider the slit disc $\\set{\\abs{z} \u003c 1} \\setminus [0, 1)$.\nSequences converging to $1/2$ from above the slit and from below it give different limits, so the Martin boundary has two points over $1/2$.\n",
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  "msc": [
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  "name": "Martin boundary",
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