The probability of an encounter of two Brownian particles before escape

TL;DR

This paper derives an explicit formula for the probability that two Brownian particles meet before one escapes a finite interval, using elliptic functions, with applications to DNA repair in confined spaces.

Contribution

It provides a novel explicit expression for the meeting probability of Brownian particles before escape, utilizing Weierstrass elliptic functions.

Findings

Explicit probability formula derived using elliptic functions

Simulation results confirm analytical accuracy

Application to DNA repair probability in confined environments

Abstract

We study the probability of two Brownian particles to meet before one of them exits a finite interval. We obtain an explicit expression for the probability as a function of the initial distance of the two particles using the Weierstrass elliptic function. We also find the law of the meeting location. Brownian simulations show the accuracy of our analysis. Finally, we discuss some applications to the probability that a double strand DNA break repairs in confined environments.