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  "body": "\n## Summary\n\nThe Martin boundary of a domain is an ideal boundary.\nIt is built from the limits of the [[Martin kernel|1DBC9FYF7M]] as the pole tends to the edge of the domain.\nEach of its points is a positive harmonic function, and every positive harmonic function on the domain is an integral of these over the boundary.\nR. S. Martin introduced it in 1941 to extend the Poisson representation from the ball to arbitrary domains.\nFor smooth or Lipschitz domains it coincides with the Euclidean boundary, but for domains with slits or cusps it can be quite different.\nIn probability it is the boundary seen by Brownian motion and by Markov chains through Doob's $h$-transforms.\n\n## Explanations\n\n### L1 human\n\nThe geometric boundary of a domain may not reflect how harmonic functions see it.\nFor a disc with a slit cut along a radius, the two sides of the slit behave like different boundary points.\nThe Martin boundary separates such points by asking which positive harmonic functions arise as limits of normalised Green's functions.\nFor the unit ball, the Martin boundary is just the unit sphere.\n",
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