On a Brownian motion conditioned to stay in an open set

TL;DR

This paper investigates the distribution of Brownian motion conditioned to remain within an open set, characterizing it via stochastic differential equations and applying findings to cluster boundaries in coalescing flows.

Contribution

It provides new characterizations of conditioned Brownian motion distributions using singular stochastic differential equations and explores applications to stochastic flow boundaries.

Findings

Distribution characterized by singular SDEs

Application to boundaries of clusters in coalescing flows

Provides a framework for conditioned Brownian motion analysis

Abstract

Distribution of a Brownian motion conditioned to start from the boundary of an open set $G$ and to stay in $G$ for a finite period of time is studied. Characterizations of such distributions in terms of certain singular stochastic differential equations are obtained. Results are applied to the study of boundaries of clusters in some coalescing stochastic flows on $\mathbb{R}.$