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  "body": "\n## Statement\n\nLet $d \\ge 2$ and let $K \\subset \\R^d$ be a compact set.\nFor $x \\in \\R^d$, let $(B_t)$ be a [[Brownian motion|22735FGJV6]] in $\\R^d$ started at $x$, with law $P^x$, and let $T_K = \\inf\\set{t \u003e 0 : B_t \\in K}$.\nThen $K$ is a [[polar set|C95QJ7YH5T]] if and only if\n$$P^x(T_K \u003c \\infty) = 0 \\quad \\text{for every } x \\in \\R^d.$$\n\n## Remarks\n\nProof idea: the hitting probability $h(x) = P^x(T_K \u003c \\infty)$ is harmonic off $K$ and is the equilibrium potential of $K$, up to regularization.\nFor $d \\ge 3$ it is the potential of the equilibrium measure of $K$.\nIt vanishes identically exactly when the Newtonian capacity of $K$ is $0$.\nFor $d = 2$ a similar argument, with logarithmic capacity and Brownian motion killed on leaving a large disc, applies.\nPolar sets are exactly the sets of capacity zero, which gives the statement.\nExample: a single point is polar for $d \\ge 2$, so planar Brownian motion started away from $0$ never hits $0$.\nA countable union of polar sets is polar, so planar Brownian motion almost surely avoids all points with rational coordinates, except possibly its starting point at time $0$.\nA line segment in $\\R^2$ is not polar, so Brownian motion hits it with positive probability.\nIn dimension $1$ the picture is different: Brownian motion hits every point almost surely, and only the empty set is polar.\n\n## Explanations\n\n### L1 human\n\nSome sets are so thin that a random path will never touch them, even though the path wanders everywhere.\nIn the plane, a single point is such a set.\nPolar sets in potential theory are exactly these sets that Brownian motion cannot see.\n\n### L2 human\n\nThis is the probabilistic description of the exceptional sets of potential theory.\nStatements that hold \"except on a polar set\", such as the boundary behaviour of Perron solutions, become statements that hold almost surely for Brownian paths.\nIt explains why polar sets are removable for bounded harmonic functions: Brownian motion never feels them.\nIn the language of Hunt processes, a set is polar for a process when the process almost surely never enters it, and this definition generalizes the classical one.\n",
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