arXiv:1011.3073v1 [math.PR] 12 Nov 2010 The greatest convex minorant of Brownian motion, meander, and bridge Jim Pitman * Nathan Ross † November 16, 2010 Abstract This article contains both a point process and a sequential description of the greatest convex minorant of Brownian motion on a finite interval. We use these descriptions to provide new analysis of various features of the convex minorant such as the set of times where the Brownian motion meets its minorant. The equivalence of the these descriptions is non-trivial, which leads to many interesting identities between quantities derived from our analysis. The sequential description can be viewed as a Markov chain for which we derive some fundamental properties. 1 Introduction The greatest convex minorant (or simply convex minorant for short) of a real- valued function (x u ,u ∈ U ) with domain U contained in the real line is the maximal convex function (c u ,u ∈ I ) defined on a closed interval I containing U with c u ≤ x u for all u ∈ U . A number of authors have provided descriptions of certain features of the convex minorant for various stochastic processes such as random walks [17], Brownian motion [9, 11, 19, 25, 28], Cauchy processes [6], Markov Processes [4], and L´ evy processes (Chapter XI of [23]). In this article, we will give two descriptions of the convex minorant of vari- ous Brownian path fragments which yield new insight into the structure of the convex minorant of a Brownian motion over a finite interval. As we shall see be- low, such a convex minorant is a piecewise linear function with infinitely many linear segments which accumulate only at the endpoints of the interval. We refer to linear segments as “faces,” the “length” of a face is as projected onto the horizontal time axis, and the slope of a face is the slope of the correspond- ing segment. We also refer to the points where the convex minorant equals the process as vertices; note that these points are also the endpoints of the linear segments. See figure 1 for illustration. * University of California at Berkeley; email pitman@stat.Berkeley.EDU; research supported in part by N.S.F. Grant DMS-0806118 † University of California at Berkeley; email ross@stat.Berkeley.EDU 1