Convergence of bi-measure R-tree and the pruning process

Abstract

In [Aldous,Pitman,1998] a tree-valued Markov chain is derived by pruning off more and more subtrees along the edges of a Galton-Watson tree. More recently, in [Abraham,Delmas,2012], a continuous analogue of the tree-valued pruning dynamics is constructed along Lévy trees. In the present paper, we provide a new topology which allows to link the discrete and the continuous dynamics by considering them as instances of the same strong Markov process with different initial conditions. We construct this pruning process on the space of so-called bi-measure trees, which are metric measure spaces with an additional pruning measure. The pruning measure is assumed to be finite on finite trees, but not necessarily locally finite. We also characterize the pruning process analytically via its Markovian generator and show that it is continuous in the initial bi-measure tree. A series of examples is given, which include the finite variance offspring case where the pruning measure is the length measure on the underlying tree.

References in corpus (9)

• A note on Gromov-Hausdorff-Prokhorov distance between (locally) compact measure spaces
• Self-similar fragmentations derived from the stable tree II: splitting at nodes
• Subtree prune and regraft: a reversible real tree-valued Markov process
• The gap between Gromov-vague and Gromov-Hausdorff-vague topology
• Self-similar fragmentations derived from the stable tree I: splitting at heights
• A limit theorem for the contour process of conditioned Galton-Watson trees
• The cut-tree of large Galton-Watson trees and the Brownian CRT
• Equivalence of Gromov-Prohorov- and Gromov's Box-Metric on the Space of Metric Measure Spaces
• Exit times for an increasing Lévy tree-valued process

Cited by in corpus (8)

• The gap between Gromov-vague and Gromov-Hausdorff-vague topology
• Invariance principle for variable speed random walks on trees
• The Aldous chain on cladograms in the diffusion limit
• Existence of mark functions in marked metric measure spaces
• Spaces of algebraic measure trees and triangulations of the circle
• Invariance principles for pruning processes of Galton-Watson trees
• Family size decomposition of genealogical trees
• Cutting down $\mathbf{p}$-trees and inhomogeneous continuum random trees