# Simultaneous boundary hitting by coupled reflected Brownian motions

## TL;DR

This paper investigates the behavior of coupled reflected Brownian motions, revealing conditions for simultaneous boundary hitting, mutual singularity of local time measures, and specific hitting scenarios in wedge domains.

## Contribution

It introduces new phenomena in coupled reflected Brownian motions, including simultaneous boundary hitting and measure singularity, expanding understanding of their boundary interactions.

## Key findings

- Uncountably many synchronized motions can hit the boundary simultaneously.
- Local time measures are mutually singular until normal vectors align.
- Mirror coupled motions can hit opposite wedge sides at different distances.

## Abstract

(i) Uncountably many synchronized reflected Brownian motions can hit the boundary of a $C^2$ domain at the same time. (ii) Measures associated to local times of two synchronized reflected Brownian motions are mutually singular until the time when the normal vectors at the reflection locations become identical. (iii) Mirror coupled reflected Brownian motions can simultaneously hit opposite sides of a wedge at different distances from the origin.