Survival probability of a Brownian motion in a planar wedge of arbitrary angle

TL;DR

This paper derives explicit formulas for the survival probability and first-passage time distribution of a Brownian motion in a planar wedge with any angle, extending previous results limited to specific angles.

Contribution

It generalizes existing results to arbitrary wedge angles, providing simple explicit expressions for survival probability and first-passage time distribution.

Findings

Explicit formulas for arbitrary wedge angles

Short-time behavior of survival probability

Extension of previous angle-specific results

Abstract

We study the survival probability and the first-passage time distribution for a Brownian motion in a planar wedge with infinite absorbing edges. We generalize existing results obtained for wedge angles of the form $\pi/n$ with $n$ a positive integer to arbitrary angles, which in particular cover the case of obtuse angles. We give explicit and simple expressions of the survival probability and the first-passage time distribution in which the difference between an arbitrary angle and a submultiple of $\pi$ is contained in three additional terms. As an application, we obtain the short time development of the survival probability in a wedge of arbitrary angle.