Extremal statistics of Brownian motion in a planar wedge

In this work, we investigate the extreme value statistics of a Brownian particle in a planar wedge prior to its first passage to the absorbing boundaries. We derive the probability distributions of both the maximum radial position M and time tm at which this extreme occurs, along with their respective expected values. Analysis of their asymptotic behavior reveals that the tails of both M and tm distributions follow power-law decays, with exponents that depend solely on the wedge angle α. For the extreme time distribution, α=π2 serves as a critical value: above this angle, the distribution exhibits a heavy-tailed power-law decay with an exponent whose absolute value is less than 2, leading to a divergent mean extreme time. In contrast, the maximum radial position becomes heavy-tailed only when α>π, where its mean diverges. In other words, for π2<α<π, the mean extreme time diverges while the mean extreme value remains finite. All theoretical results are validated by numerical simulations.