# Hitting distributions of planar Brownian motion

## TL;DR

This paper derives formulas for the joint distribution of hitting times and locations of planar Brownian motion hitting the negative horizontal axis, including cases with drift, advancing understanding of stochastic boundary hitting behaviors.

## Contribution

It provides new explicit formulas for Green functions and conditional distributions for planar Brownian motion hitting a boundary, including drifted cases.

## Key findings

- Derived formulas for Green functions and hitting distributions.
- Extended results to Brownian motion with negative drift.
- Enhanced understanding of boundary hitting behavior in 2D Brownian motion.

## Abstract

The purpose of the paper is to find the joint distribution of the hitting time and place of two-dimensional Brownian motion hitting the negative horizontal axis. We provide various formulas for Green functions as well as for the conditional distributions. In the end we treat an analogous problem for the case when vertical component has a negative drift.