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.
Article PDF
Similar content being viewed by others
References
Aldous, D., Fill, J.: Reversible Markov chains and random walks on graphs. Monograph in preparation. http://www.stat.berkeley.edu/~aldous/RWG/book.html (1994–2002)
Bıyıkoŭlu T. (2003). A discrete nodal domain theorem for trees. Linear Algebra Appl. 360: 197–205
Burdzy, K.: Neumann eigenfunctions and Brownian couplings. https://digital.lib.washington.edu/dspace/bitstream/1773/2130/1/iwpt.pdf (2007)
Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in analysis (Papers dedicated to Salomon Bochner, 1969), pp. 195–199. Princeton University Press, Princeton (1970)
Chen M.-F. and Wang F.-Y. (1997). Estimation of spectral gap for elliptic operators. Trans. Amer. Math. Soc. 349(3): 1239–1267
Chen M. (1999). Analytic proof of dual variational formula for the first eigenvalue in dimension one. Sci. China Ser. A 42(8): 805–815
Chen M. (2001). Variational formulas and approximation theorems for the first eigenvalue in dimension one. Sci. China Ser. A 44(4): 409–418
Courant R. and Hilbert D. (1953). Methods of mathematical physics, vol. I. Interscience Publishers Inc., New York
Del Moral, P., Miclo, L.: Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM Probab. Stat. 7, 171–208 (electronic) (2003)
Diaconis P. and Allen Fill J. (1990). Strong stationary times via a new form of duality. Ann. Probab. 18(4): 1483–1522
Fomin S. and Zelevinsky A. (2000). Total positivity: tests and parametrizations. Math. Intelligencer 22(1): 23–33
Friedman J. (1993). Some geometric aspects of graphs and their eigenfunctions. Duke Math. J. 69(3): 487–525
Fukushima, M., Ōshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes, volume 19 of de Gruyter Studies in Mathematics. Walter de Gruyter, Berlin (1994)
Gantmacher F.R. and Krein M.G. (1937). Sur les matrices complétement non négatives et oscillatoires. Compositio Math. 4: 445–470
Karlin S. and McGregor J.L. (1957). The differential equations of birth-and-death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85: 489–546
Karlin S. (1968). Total positivity, vol. I. Stanford University Press, Stanford
Lawler G.F. and Sokal A.D. (1988). Bounds on the L 2 spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Amer. Math. Soc. 309(2): 557–580
Miclo, L.: Monotonicité des fonctions extrémales pour les inégalités de type Sobolev logarithmiques en dimension 1. http://hal.ccsd.cnrs.fr/ccsd-00019571 (2005)
Schoenberg I. (1930). Über variationsvermindernde lineare transformationen. Math. Z. 32(1): 321–328
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-007-0115-9
Keywords
- Birth and death processes
- Markov processes on trees
- Eigendecomposition of generators
- Dirichlet eigenproblems
- Isospectral partition
- Nodal domains
- Cheeger inequalities
- Spectral gap