Mathematics – Probability
Scientific paper
2011-11-24
Mathematics
Probability
5 pages. In the new version, as an application, a short proof of the following fact is given: The R-tree coded by an excursion
Scientific paper
Stochastic processes with values in the space of metric measure spaces (complete separable metric spaces equipped with a probability measure) are becoming more and more important in probability theory, especially for the modelling of evolutionary systems, where at each time the whole phylogenetic tree is considered. Greven, Pfaffelhuber and Winter introduced the Gromov-Prohorov metric d_{GPW} on the space of metric measure spaces and showed that it induces the Gromov-weak topology. They also conjectured that this topology coincides with the topology induced by Gromov's Box_1 metric. In this note, we show that this is indeed true, and the metrics are even bi-Lipschitz equivalent. More precisely, d_{GPW}= 1/2 Box_{1/2} and hence d_{GPW} <= Box_1 <= 2d_{GPW}. As an application, we give an easy proof of the known fact that the map associating to an excursion the coded R-tree is (Lipschitz)-continuous.
Löhr Wolfgang
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