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  "body": "\n## Statement\n\nLet $d \\ge 2$ and let $D \\subset \\R^d$ be a bounded [[domain|0Q4CBD3HX0]] whose boundary points are all regular for the Dirichlet problem.\nFor $x \\in D$, let $\\omega_x$ be the [[harmonic measure|CFD7BPM51W]] of $D$ at $x$.\nIt is the [[Borel probability measure|DYR1QFNGNJ]] on $\\partial D$ such that $\\int \\varphi \\, d\\omega_x = u_\\varphi(x)$ for every continuous $\\varphi$ on $\\partial D$.\nHere $u_\\varphi$ is the solution of the Dirichlet problem with boundary data $\\varphi$.\nLet $(B_t)$ be a [[Brownian motion|22735FGJV6]] in $\\R^d$ started at $x$, with law $P^x$, and let $\\tau = \\inf\\set{t \u003e 0 : B_t \\notin D}$.\nThen for every Borel set $A \\subseteq \\partial D$,\n$$P^x(B_\\tau \\in A) = \\omega_x(A).$$\n\n## Remarks\n\nProof idea: by [[Kakutani's theorem|CCEKM4241P]], $E^x[\\varphi(B_\\tau)]$ is the solution of the Dirichlet problem with data $\\varphi$, for each continuous $\\varphi$.\nSo the law of $B_\\tau$ and $\\omega_x$ give the same integral to each continuous function on the compact set $\\partial D$, and the two Borel measures are equal.\nExample: for the ball $\\abs{y} \u003c 1$ in $\\R^d$, the exit point from $x$ has density, with respect to normalized surface measure, equal to the [[Poisson kernel|TCYSWCTYYM]] $(1 - \\abs{x}^2)/\\abs{x - \\zeta}^d$.\nFrom the centre $x = 0$ this density is $1$, so the exit point is uniform on the sphere.\nWithout regularity, the identity still holds when $\\omega_x$ is the harmonic measure defined through Perron solutions.\nThe set of irregular boundary points is then polar, and it gets no mass.\n\n## Explanations\n\n### L1 human\n\nRelease a particle at a point inside a region and let it move at random until it reaches the boundary.\nThe harmonic measure of a piece of the boundary is the chance that the particle first arrives there.\nPieces close to the start point, or wide open to it, get more probability; pieces hidden in narrow fjords get very little.\n\n### L2 human\n\nThe probabilistic picture explains many properties of harmonic measure at once.\nIt is a probability measure, it depends harmonically on the starting point, and it is monotone in the domain.\nIn the plane, Brownian motion is conformally invariant up to a time change, which explains the conformal invariance of harmonic measure.\nEstimates such as Beurling's projection theorem have natural proofs and interpretations in this language.\n",
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        "locator": "The harmonic measure of a diffusion",
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  "name": "Harmonic measure as exit distribution",
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