# Probability of winding number in 2D Brownian motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau \leq 1]$?

• This is an interesting question, what have you tried to far? – nullUser Oct 22 '13 at 22:23
• @nullUser I was trying to model the problem with a differential equation. For example, the probability distribution of $B_t$ satisfies the heat equation and one can certainly solve this equation in polars and impose appropriate initial and boundary conditions. However, I can't link this probability distribution and the distribution of $\tau$. I have also tried to derive the distribution of $\theta$ or $\tau$ with martingale theory, but have not succeeded. – rodms Oct 22 '13 at 22:31