Random Walk Winding Number — Punctured Plane

How many times does a Brownian path wind around the origin?

10
6
3

Winding Number of Brownian Motion (Spitzer 1958)

Define θ(t) = arg(B(t)) — the angle of a 2D Brownian path around the origin. The winding number W(t) = θ(t)/(2π) counts net revolutions. Spitzer's theorem:

2·θ(t)/log(t) → Cauchy(0,1) as t → ∞

So θ(t) ~ π·Cauchy · log(t)/2. The typical winding grows as log(t) — extremely slowly. The walk explores all winding classes, but the distribution has heavy tails (Cauchy, not Gaussian). Related to the logarithmic potential of 2D BM and the topology of π₁(ℝ²\{0}) = ℤ.