Open Access
2024 Extended Lévy’s theorem for a two-sided reflection
Benjamin Housley
Author Affiliations +
Electron. Commun. Probab. 29: 1-12 (2024). DOI: 10.1214/24-ECP576


We aim to set forth an extension of the result found in paper [6], which finds an explicit realisation of a reflecting Brownian motion with drift μ, started at x, reflecting above zero, and its local time at zero. In this paper we find a corresponding realisation for a reflecting Brownian motion with drift μ, started at x, reflected both above zero and below one, along with a corresponding expression in terms of associated local times, namely as the difference between the local time at zero and the local time at one.

Funding Statement

Supported by the Institute of Mathematical Statistics (IMS) and the Bernoulli Society.


The author thanks Goran Peskir for his useful discussions and comments about this work, and also thanks the anonymous referees for their careful reading and comments that have improved the paper.


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Benjamin Housley. "Extended Lévy’s theorem for a two-sided reflection." Electron. Commun. Probab. 29 1 - 12, 2024.


Received: 16 May 2023; Accepted: 11 February 2024; Published: 2024
First available in Project Euclid: 20 March 2024

Digital Object Identifier: 10.1214/24-ECP576

Primary: 60J50 , 60J55 , 60J65
Secondary: 60J60

Keywords: diffusion process , Lévy’s theorem , Local time , normal reflection , reflecting Brownian motion with drift , Skorokhod map , two-sided reflecting Brownian motion

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