On the duality between jump processes on ultrametric spaces and random walks on trees

Wolfgang Woess

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arXiv · 1211.7216

On the duality between jump processes on ultrametric spaces and random walks on trees

Wolfgang Woess

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Abstract

The purpose of these notes is to clarify the duality between a natural class of jump processes on compact ultrametric spaces - studied in current work of Bendikov, Girgor'yan and Pittet - and nearest neighbour walks on trees. Processes of this type have appeared in recent work of Kigami. Every compact ultrametric space arises as the boundary of a locally finite tree. The duality arises via the Dirichlet forms: one on the tree associated with a random walk and the other on the boundary of the tree, which is given in terms of the Na\"im kernel. Here, it is explained that up to a linear time change by a unique constant, there is a one-to-one correspondence between the above processes and Dirichlet regular random walks.

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Wolfgang Woess. 2012-11-30. On the duality between jump processes on ultrametric spaces and random walks on trees. https://arxiv.org/abs/1211.7216

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