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On eigenfunctions of Markov processes on trees
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  • Published: 27 November 2007

On eigenfunctions of Markov processes on trees

  • Laurent Miclo1 

Probability Theory and Related Fields volume 142, pages 561–594 (2008)Cite this article

Abstract

We begin by studying the eigenvectors associated to irreducible finite birth and death processes, showing that the i nontrivial eigenvector φ i admits a succession of i decreasing or increasing stages, each of them crossing zero. Imbedding naturally the finite state space into a continuous segment, one can unequivocally define the zeros of φ i , which are interlaced with those of φ i+1. These kind of results are deduced from a general investigation of minimax multi-sets Dirichlet eigenproblems, which leads to a direct construction of the eigenvectors associated to birth and death processes. This approach can be generically extended to eigenvectors of Markov processes living on trees. This enables to reinterpret the eigenvalues and the eigenvectors in terms of the previous Dirichlet eigenproblems and a more general conjecture is presented about related higher order Cheeger inequalities. Finally, we carefully study the geometric structure of the eigenspace associated to the spectral gap on trees.

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Authors and Affiliations

  1. Laboratoire d’Analyse, Topologie, Probabilités, Centre de Mathématiques et Informatique, Université de Provence, 39, rue Frédéric Joliot-Curie, 13453, Marseille cedex 13, France

    Laurent Miclo

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  1. Laurent Miclo
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Correspondence to Laurent Miclo.

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Miclo, L. On eigenfunctions of Markov processes on trees. Probab. Theory Relat. Fields 142, 561–594 (2008). https://doi.org/10.1007/s00440-007-0115-9

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  • Received: 11 May 2007

  • Revised: 17 October 2007

  • Published: 27 November 2007

  • Issue Date: November 2008

  • DOI: https://doi.org/10.1007/s00440-007-0115-9

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Keywords

  • Birth and death processes
  • Markov processes on trees
  • Eigendecomposition of generators
  • Dirichlet eigenproblems
  • Isospectral partition
  • Nodal domains
  • Cheeger inequalities
  • Spectral gap

Mathematical Subject Classification (2000)

  • Primary: 60J80
  • Secondary: 15A18
  • 49R50
  • 26A48
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