# A topology for labelled metric spaces, application to s-compact random genealogical trees

Abstract : In this thesis, we develop a new space for the study of measured labelled metric spaces, ultimately designed to represent genealogical trees with a root at generation minus infinity. The time in the genealogical tree is represented by a 1-Lipschitz label function. We define the notion of S-compact measured labelled metric space, that is a metric space E equipped with a measure nu and a 1-Lipschitz label function from E to R, with the additional condition that each slice (the set of points with labels in a compact of R) must be compact and have finite measure. On the space XS of measured labelled metric spaces (up to isometry), we define a distance dLGHP by comparing the slices and study the resulting metric space, which we find to be Polish.We proceed with the study of the set T of all elements of XS that are real tree in which the label function decreases at rate 1 when we go toward the root" (which can be infinitely far). Each possible value of the label function corresponds to a generation in the genealogical tree. We prove that (T, dLGHP) is Polish as well. We define a number of measurable operation on T, including a way to randomly graft a forest on a tree. We use this operation to build a particular random tree generalizing Aldous' Brownian motion conditioned on its local time
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• HAL Id : tel-02914733, version 1

### Citation

Gustave Emprin. A topology for labelled metric spaces, application to s-compact random genealogical trees. Probability [math.PR]. Université Paris-Est, 2019. English. ⟨NNT : 2019PESC1032⟩. ⟨tel-02914733⟩

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