Brownian motion conditioned to stay in a cone

TL;DR

This paper extends a known limit theorem for Brownian motion conditioned to stay positive to the case of multidimensional Brownian motion constrained within a smooth convex cone, broadening understanding of conditioned stochastic processes.

Contribution

It generalizes the limit theorem for Brownian meander from one dimension to multiple dimensions within convex cones, providing new insights into multidimensional stochastic conditioning.

Findings

Extended the limit theorem to multidimensional Brownian motion in convex cones

Established weak convergence of conditioned Brownian motion in higher dimensions

Broadened the theoretical understanding of stochastic processes in constrained domains

Abstract

A result of R. Durrett, D. Iglehart and D. Miller states that Brownian meander is Brownian motion conditioned to stay positive for a unit of time, in the sense that it is the weak limit, as $x$ goes to 0, of Brownian motion started at $x>0$ and conditioned to stay positive for a unit of time. We extend this limit theorem to the case of multidimensional Brownian motion conditioned to stay in a smooth convex cone.