On eigenfunctions of Markov processes on trees

Laurent Miclo (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

We begin by studying the eigenvectors associated to irreducible finite birth and death processes, showing that the $i^{\mathrm{th}}$ nontrivial eigenvector $\varphi_i$ 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 $\varphi_i$, which are interlaced with those of $\varphi_{i+1}$. 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.

Mots clés: birth and death processes; Markov processes on trees; eigendecomposition of generators; Dirichlet eigenproblems; isospectral partition; nodal domains; Cheeger inequalities; spectral gap.; spectral gap

Domaines: Probabilités [math.PR]

Laurent Miclo. On eigenfunctions of Markov processes on trees. Probability Theory and Related Fields, 2008, 142 (3-4), pp.561-594. ⟨hal-00147865⟩