Mathematics – Probability
Scientific paper
2006-04-05
Mathematics
Probability
Paper, 50 pages, 4 figures
Scientific paper
This paper is a detailled study of the coding of real trees by real valued functions that is motivated by probabilistic problems related to continuum random trees. Indeed it is known since the works of Aldous (1993) and Le Gall (1991) that a continuous non-negative function $h$ on $[0,1]$ such that $h(0)=0$ can be seen as the contour process of a compact real tree. This particular coding of a compact real tree provides additional structures, namely a root that is the vertex corresponding to $0\in [0,1]$, a linear order inherited from the usual order on $[0,1]$ and a measure induced by the Lebesgue measure on $[0,1]$; of course, the root, the linear order and the measure obtained by such a coding have to satisfy some compatibility conditions. In this paper, we prove that any compact real tree equipped with a root, a linear order and a measure that are compatible can be encoded by a non-negative function $h$ defined on a finite interval $[0, M]$, that is assumed to be left-continuous with right-limit, without positive jump and such that $h(0+)=h(0)=0$. Moreover, this function is unique if we assume that the exploration of the tree induced by such a coding backtracks as less as possible. We also prove that a measure-change on the tree corresponds to a re-parametrization of the coding function. In addition, we describe several path-properties of the coding function in terms of the metric properties of the real tree.
No associations
LandOfFree
The coding of compact real trees by real valued functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The coding of compact real trees by real valued functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The coding of compact real trees by real valued functions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-418806