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The exponential map is a conformal bijection from the stripto the wedge containing the starting point $1$. The restriction $\alpha +\beta <2\pi$ makes it injective. A planar Brownian motion $U+iV$ started at zero in this strip maps, after the conformal Brownian clock, to the required Brownian motion in the wedge.

Let $\sigma$ be the first time $V$ exits $(-\alpha ,\beta )$. It is finite almost surely, and $V_{t\wedge \sigma }$ is a bounded martingale. The optional stopping theorem givesThe strip exit maps to the corresponding bounding ray. The exponential clock is finite at $\sigma$, because its integrand is continuous on the compact time interval $[0,\sigma ]$. Thus it does not remove or reverse the exit event. We obtainThis is the Brownian wedge-exit probability obtained directly from the strip, using conformal invariance of planar Brownian motion.

Substitute $R=r^{\pi /\alpha }$ from part (a) into part (c). The Brownian wedge-exit probability isIt tends to $1$ as $r\downarrow 1$, and is asymptotic to $(4/\pi )r^{-\pi /\alpha }$ as $r\to \infty$. The exponent reflects how the conformal map stretches wedge angles; wider wedges give slower decay.