Solution

Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 29 / 2 / d / Solution

{c}
{title2=$p_\alpha(r)=\frac4\pi\arctan(r^{-\pi/\alpha})$}

= Wedge survival probability
{synonym}

A <planar Brownian motion> starting at $1$ exits the disk of radius $r>1$ before leaving a wedge of opening $\alpha\in(0,2\pi]$, centered on the positive axis, with probability $(4/\pi)\arctan(r^{-\pi/\alpha})$. The <power-map reduction for Brownian exit from a wedge>, the <reflection identity for Brownian exit from a half-disc> and the <Möbius calculation of circular Brownian exit> give the formula. Its large-radius decay exponent is $\pi/\alpha$.

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 Incoming links: Brownian wedge-exit probability