On the exit time from a cone for brownian motion with drift

Rodolphe Garbit,Kilian Raschel

doi:10.1214/ejp.v19-3169

Rodolphe Garbit, Kilian Raschel

Open Access

https://doi.org/10.1214/ejp.v19-3169

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Abstract

We investigate the tail distribution of the first exit time of Brownian motion with drift from a cone and find its exact asymptotics for a large class of cones. Our results show in particular that its exponential decreasing rate is a function of the distance between the drift and the cone, whereas the polynomial part in the asymptotics depends on the position of the drift with respect to the cone and its polar cone, and reflects the local geometry of the cone at the points that minimize the distance to the drift.

Highlights

Let Bt be a d-dimensional Brownian motion with drift a ∈ Rd
In this article we study the probability for the Brownian motion started at x not to exit C before time t, namely, (1)
In [24], Spitzer considered the case d = 2 and obtained an explicit expression for the probability (1) for any two-dimensional cone. He introduced the winding number process θt = arg Bt

Summary

Introduction

Let Bt be a d-dimensional Brownian motion with drift a ∈ Rd. For any cone C ⊂ Rd, define the first exit time τC = inf{t > 0 : Bt ∈/ C}. In [24], Spitzer considered the case d = 2 and obtained an explicit expression for the probability (1) for any two-dimensional cone He introduced the winding number process θt = arg Bt (in dimension d = 2, the Brownian motion does not exit a given cone before time t if and only if θt stays in some interval). In [1], Banuelos and Smits refined the results of DeBlassie [8, 9]: they considered more general cones, and obtained a quite tractable expression for the heat kernel (the transition densities for the Brownian motion in C killed on the boundary), and for (1) We conclude this part by mentioning the work [10], in which Doumerc and O’Connell found a formula for the distribution of the first exit time of Brownian motion from a fundamental region associated with a finite reflection group.

Assumptions on the cone and statements of results

The example of two-dimensional Brownian motion in cones

Preliminary results

Precise statements and proofs of the theorems A–F