Random walks on Bratteli diagrams

Jean Renault

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arXiv · 1704.06990

Random walks on Bratteli diagrams

Jean Renault

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Abstract

In the eighties, A. Connes and E. J. Woods made a connection between hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks. The present paper explains this connection and gives a detailed proof of two theorems quoted there: the construction of a large class of states on a hyperfinite von Neumann algebra (due to A. Connes) and the ergodic decomposition of a Markov measure via harmonic functions (a classical result in probability theory). The crux of the first theorem is a model for conditional expectations on finite dimensional C*-algebras. The proof of the second theorem hinges on the notion of cotransition probability.

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Jean Renault. 2017-04-23. Random walks on Bratteli diagrams. https://arxiv.org/abs/1704.06990

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