Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

2009/05/18 by Alessandra Faggionato, Faggionato, A.
Mathematics · #34B24 #60K37 #82C44 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Spectral Theory (math.SP) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0905.2900

openalex publication_date 2009/05/18 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

We consider a family X⁽ⁿ⁾, n ∈ \bbN₊, of continuous-time nearest-neighbor random walks on the one dimensional lattice Z. We reduce the spectral analysis of the Markov generator of X⁽ⁿ⁾ with Dirichlet conditions outside (0,n) to the analogous problem for a suitable generalized second order differential operator -D_mn D_x, with Dirichlet conditions outside a given interval. If the measures dm_n weakly converge to some measure dm_*, we prove a limit theorem for the eigenvalues and eigenfunctions of -D_mnD_x to the corresponding spectral quantities of -D_{m_*} D_x. As second result, we prove the Dirichlet-Neumann bracketing for the operators -D_m D_x and, as a consequence, we establish lower and upper bounds for the asymptotic annealed eigenvalue counting functions in the case that m is a self--similar stochastic process. Finally, we apply the above results to investigate the spectral structure of some classes of subdiffusive random trap and barrier models coming from one-dimensional physics.

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