On Conditioning Brownian Particles to Coalesce

TL;DR

This paper introduces a new way to condition Brownian particles on rare events, showing that their behavior after coalescence matches a modified flow model, advancing understanding of stochastic coalescence processes.

Contribution

It defines a conditional distribution for Brownian particles given a zero-probability event and links it to a modified Arratia flow, providing a novel theoretical framework.

Findings

Conditional distribution matches the modified Arratia flow law

Coalescence behavior is characterized under the new conditioning

Provides a rigorous mathematical foundation for conditioned Brownian coalescence

Abstract

We introduce the notion of a conditional distribution to a zero-probability event in a given direction of approximation, and prove that the conditional distribution of a family of independent Brownian particles to the event that their paths coalesce after the meeting coincides with the law of a modified massive Arratia flow, defined in [arXiv:1408.0628].