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Stochastic and Infinite Dimensional Analysis pp 205–226Cite as

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Stochastic Processes on Ends of Tree and Dirichlet Forms

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Abstract

We present main ideas and compare two constructions of stochastic processes on the ends (leaves) of the trees with varying numbers of edges at the nods. In one of them the trees are represented by spaces of numerical sequences and the processes are obtained by solving a class of Chapman-Kolmogorov Equations. In the other the trees are described by the set of nodes and edges. To each node there is naturally associated a finite dimensional function space and the Dirichlet form on it. Having a class of Dirichlet forms at the nodes one can under certain conditions build a Dirichlet form on L 2 space of funcions on the ends of the trees. We show that the state spaces of two approaches are homeomorphic but the second yields larger class of processes.

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Author information

Authors and Affiliations

  1. Institute of Pysics, Opole University, Opole, Poland

    Witold Karwowski

Authors
  1. Witold Karwowski
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Correspondence to Witold Karwowski .

Editor information

Editors and Affiliations

  1. Research Center for Theoretical Physics Central Visayan Institute Foundation, Jagna, Bohol, Philippines

    Christopher C. Bernido

  2. Research Center for Theoretical Physics Central Visayan Institute Foundation, Jagna, Bohol, Philippines

    Maria Victoria Carpio-Bernido

  3. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Martin Grothaus

  4. Department of Mathematics and Physics, University of Reading, Reading, United Kingdom

    Tobias Kuna

  5. Departamento de Ciências e Tecnologia, Universidade Aberta, Lisbon, Portugal

    Maria João Oliveira

  6. Universidade da Madeira Faculdade de Ciências Exatas e da Engenharia, Funchal, Madeira, Portugal

    José Luís da Silva

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Karwowski, W. (2016). Stochastic Processes on Ends of Tree and Dirichlet Forms. In: Bernido, C., Carpio-Bernido, M., Grothaus, M., Kuna, T., Oliveira, M., da Silva, J. (eds) Stochastic and Infinite Dimensional Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-07245-6_11

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