2012/11/30 by Wolfgang Woess, Woess, Wolfgang
Mathematics · #05C05 #31C05 #60G50 #60J50 #FOS: Mathematics #Probability (math.PR) #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics #advanced mathematical theories #math.PR #msc:05C05 #msc:31C05 #msc:60G50 #msc:60J50
paper · pdf · doi:10.48550/arxiv.1211.7216
arxiv created 2012/11/30 · openalex publication_date 2012/11/30 · arxiv updated 2012/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Naïm 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.