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

   
@article{EJP1831,
	author = {Alessandra Faggionato},
	title = {Spectral analysis of 1D nearest-neighbor random walks  and applications to subdiffusive trap and barrier models},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {17},
	year = {2012},
	keywords = {random walk ; generalized differential operator ; Sturm-Liouville theory ; random trap model ; random barrier model ; self--similarity ; Dirichlet--Neumann bracketing},
	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.
},
	pages = {no. 15, 1-36},
	issn = {1083-6489},
	doi = {10.1214/EJP.v17-1831},    
        url = {http://ejp.ejpecp.org/article/view/1831}}