一维最近邻随机游动的谱分析及其在次扩散陷阱和势垒模型中的应用
Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models
摘要：
考虑一维格Z上连续时间最近邻随机游动族X^{（n）}，n \in \bbN_+.本文将X^{（n）}在（0，n）外Dirichlet条件下的Markov生成元的谱分析问题归结为一个适当的广义二阶微分算子D_{m_n} D_x在给定区间外Dirichlet条件下的类似问题.若测度dm_n弱收敛于某测度dm_*，则证明了- D_{m_n} D_x的特征值和特征函数到- D_{m_*} D_x的谱量的一个极限定理.作为第二个结果，我们证明了算子-D_mD_x的Dirichlet-Neumann括号，并由此建立了当m是自相似随机过程时，渐近退火本征值计数函数的上下界。最后，我们应用上述结果研究了一维物理中的几类次扩散随机陷阱和势垒模型的谱结构。
英文摘要：
We consider a family X^{(n)}, n \in \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_{m_n} 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_{m_n}D_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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
