## Electronic Journal of Probability

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

Alessandra Faggionato

#### Abstract

We consider a   sequence  $X^{(n)}$, $n \geq 1$,   of continuous-time nearest-neighbor random walks on the one dimensional lattice $\mathbb{Z}$.  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 $-D_{m_n} D_x$, with Dirichlet conditions outside a giveninterval. If  the measures $dm_n$ weakly converge to some measure $dm_\infty$,  we prove a limit theorem for the eigenvalues and eigenfunctions of $-D_{m_n}D_x$ to the corresponding spectral quantities of $-D_{m_\infty} D_x$.  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.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 15, 36 pp.

Dates
Accepted: 24 February 2012
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465062337

Digital Object Identifier
doi:10.1214/EJP.v17-1831

Mathematical Reviews number (MathSciNet)
MR2892328

Zentralblatt MATH identifier
1248.60105

Rights

#### Citation

Faggionato, Alessandra. Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models. Electron. J. Probab. 17 (2012), paper no. 15, 36 pp. doi:10.1214/EJP.v17-1831. https://projecteuclid.org/euclid.ejp/1465062337

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