polar sets and capacity

Which sets does Brownian motion hit, and which does it miss entirely? A set is POLAR if, started from any point outside it, Brownian motion never hits it (with probability one). Capacity is the quantitative gauge of 'hittability' from potential theory: a set has positive capacity exactly when it is hit with positive probability, and zero capacity exactly when it is polar. Together they explain the dramatic dimension-dependence of Brownian motion — why a point is invisible in the plane but a curve is not, and why recurrence flips to transience at d = 3.

The recurrence/transience dichotomy: Brownian motion is neighbourhood-recurrent in d = 1 and d = 2 (it returns to every neighbourhood of every point infinitely often) but TRANSIENT in d >= 3 (|B_t| -> infinity, so it eventually leaves every bounded set forever). Yet even in d = 2 single POINTS are polar: with probability one, planar Brownian motion never returns to its exact starting point (it is neighbourhood-recurrent but point-recurrent only in d = 1). In d >= 2 every singleton is polar; more refined, a set A is polar if and only if it has zero capacity. The relevant capacity is the energy-based (Newtonian/logarithmic) capacity: cap(A) is the reciprocal of the minimal energy integral (over R^d x R^d of the Green/Riesz kernel against a probability measure supported on A), and a compact set is hit by Brownian motion from outside if and only if it has a probability measure of finite energy — Kakutani's theorem. There is a sharp DIMENSION test: a compact set of Hausdorff dimension s is a.s. hit by d-dimensional Brownian motion if s > d - 2 and a.s. missed if s < d - 2 (the borderline needs capacity). So in d = 3 a curve (dimension 1 = d - 2) is critical, surfaces are hit, points (dimension 0) are missed.

Why it matters: capacity is the bridge from probability to electrostatics — the equilibrium (harmonic) measure on a conductor is the exit/hitting distribution, and physical capacitance equals Wiener capacity up to constants. Polarity governs whether boundary data 'on a thin set' is seen by the Dirichlet problem, whether a process can be killed on a target, and the regularity of boundary points (Wiener's criterion is a capacity series). The honest content: polar is about EXACT hitting, not approximate — planar Brownian motion comes arbitrarily close to its start infinitely often (neighbourhood-recurrent) yet hits the exact point with probability zero; do not conflate 'gets near' with 'hits'. And capacity, not Lebesgue measure or even Hausdorff dimension alone, is the precise determinant of hitting at the critical dimension.