Brownian motion in a truncated Weyl chamber

TL;DR

This paper analyzes the probability that multidimensional Brownian motion remains within a growing truncated Weyl chamber, identifying decay regimes and deriving large deviation principles as the dimension increases.

Contribution

It introduces explicit eigenvalue expansions for transition probabilities and characterizes decay regimes based on growth speed, advancing understanding of Brownian motion in complex domains.

Findings

Decay regimes depend on growth speed: polynomial, stretched-exponential, exponential.

Explicit eigenvalue expansion for transition probabilities is derived.

Large deviation principles for empirical measures are established as dimension grows.

Abstract

We examine the non-exit probability of a multidimensional Brownian motion from a growing truncated Weyl chamber. Different regimes are identified according to the growth speed, ranging from polynomial decay over stretched-exponential to exponential decay. Furthermore we derive associated large deviation principles for the empirical measure of the properly rescaled and transformed Brownian motion as the dimension grows to infinity. Our main tool is an explicit eigenvalue expansion for the transition probabilities before exiting the truncated Weyl chamber.