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 . In particular, the one-dimensional distribution of is that of , where is a 5-dimensional Bessel process with . The process shares a number of other properties with , and we conjecture that it may have the distribution of . 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.
"Markovian structure in the concave majorant of Brownian motion." Electron. J. Probab. 27 1 - 21, 2022. https://doi.org/10.1214/22-EJP769