Open Access
2022 Markovian structure in the concave majorant of Brownian motion
Mehdi Ouaki, Jim Pitman
Author Affiliations +
Electron. J. Probab. 27: 1-21 (2022). DOI: 10.1214/22-EJP769


The purpose of this paper is to highlight some hidden Markovian structure of the concave majorant of the Brownian motion. Several distributional identities are implied by the joint law of a standard one-dimensional Brownian motion B and its almost surely unique concave majorant K on [0,). In particular, the one-dimensional distribution of 2KtBt is that of R5(t), where R5 is a 5-dimensional Bessel process with R5(0)=0. The process 2KB shares a number of other properties with R5, and we conjecture that it may have the distribution of R5. We also describe the distribution of the convex minorant of a three-dimensional Bessel process with drift.


The authors would like to thank an anonymous referee for his insightful comments and for allowing the use of his picture in Figure 1.


Download Citation

Mehdi Ouaki. Jim Pitman. "Markovian structure in the concave majorant of Brownian motion." Electron. J. Probab. 27 1 - 21, 2022.


Received: 24 May 2021; Accepted: 25 March 2022; Published: 2022
First available in Project Euclid: 6 May 2022

MathSciNet: MR4417199
zbMATH: 1492.60230
Digital Object Identifier: 10.1214/22-EJP769

Primary: 60G51 , 60G55 , 60J65

Keywords: Brownian motion , convex minorant , Path decomposition

Vol.27 • 2022
Back to Top