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


 
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1. Title Title of document Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models
 
2. Creator Author's name, affiliation, country Alessandra Faggionato; University La Sapienza, Rome; Italy
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) random walk ; generalized differential operator ; Sturm-Liouville theory ; random trap model ; random barrier model ; self--similarity ; Dirichlet--Neumann bracketing
 
3. Subject Subject classification 60K37 ; 82C44 ; 34B24
 
4. Description 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.
 
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7. Date (YYYY-MM-DD) 2012-02-24
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/1831
 
10. Identifier Digital Object Identifier 10.1214/EJP.v17-1831
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 17
 
12. Language English=en en
 
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