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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a domain that has a [[Green's function|M36SD56MWQ]] $G \\colon \\Omega \\times \\Omega \\to (0, +\\infty]$.\nThis holds for every domain when $n \\ge 3$, and for a plane domain whose complement is not a polar set.\nFix a reference point $x_0 \\in \\Omega$.\nThe *Martin kernel* of $\\Omega$ with reference point $x_0$ is the function\n$$M(x, y) = \\frac{G(x, y)}{G(x_0, y)} \\qquad \\text{for } x \\in \\Omega,\\ y \\in \\Omega \\setminus \\set{x_0}.$$\n\n## Remarks\n\nFor fixed $y$, the function $M(\\cdot, y)$ is positive and [[harmonic|0DEB0JMN0W]] on $\\Omega \\setminus \\set{y}$, and $M(x_0, y) = 1$.\nBy [[Harnack's inequality|YNH832D50W]], when $y$ stays away from a compact set $K \\ni x_0$, the functions $M(\\cdot, y)$ are uniformly bounded on $K$.\nLet $(y_k)$ be a sequence that leaves each compact subset of $\\Omega$.\nThen a subsequence of $M(\\cdot, y_k)$ converges locally uniformly to a positive harmonic function $h$ with $h(x_0) = 1$.\nExample: let $\\Omega$ be the unit ball of $\\R^n$ and $x_0 = 0$.\nIf $y_k \\to \\zeta$ with $\\abs{\\zeta} = 1$, then $M(x, y_k) \\to (1 - \\abs{x}^2) / \\abs{x - \\zeta}^n$, the [[Poisson kernel|TCYSWCTYYM]] for the normalised surface measure.\nA change of reference point $x_0$ multiplies $M(\\cdot, y)$ by the factor $G(x_0, y) / G(x_0', y)$.\n",
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