Exit problems associated with affine reflection groups

TL;DR

This paper derives formulas for the distribution of the first exit time of Brownian motion from alcoves of affine Weyl groups, extending to other Markov processes and confirming the Hot Spots conjecture for alcoves.

Contribution

It provides a compact Pfaffian formula for exit time distributions and explicit expected exit times in type A, also analyzing eigenfunctions of Laplacians on alcoves.

Findings

Pfaffian formulas for exit time distributions

Expected exit times in type A case

Confirmation of the Hot Spots conjecture for alcoves

Abstract

We obtain a formula for the distribution of the first exit time of Brownian motion from the alcove of an affine Weyl group. In most cases the formula is expressed compactly, in terms of Pfaffians. Expected exit times are derived in the type \~A case. The results extend to other Markov processes. We also give formulas for the real eigenfunctions of the Dirichlet and Neumann Laplacians on alcoves, observing that the `Hot Spots' conjecture of J. Rauch is true for alcoves.