Estimation of the smallest eigenvalue in fractional escape problems: Semi-analytics and fits

Continuous time random walks with heavy tailed distributions of waiting times and jump lengths lead to situations when evolution of a probability density of finding a particle at given point at given time is described by the bi-fractional Smoluchowski-Fokker-Planck equation. A power-law distribution of waiting times results in very general properties of a survival probability which in turn can be used to estimate eigenvalues of some fractional operators. Here, the problem of numerical estimation of the smallest eigenvalues is discussed for the two generic problems: escape from a finite interval and the Kramers problem of escape from a potential well. We discuss both how to numerically obtain the (effective) smallest eigenvalue of the problem, and how it can be used in numerically assessing other important characteristics of the processes. (C) 2014 Elsevier B.V. All rights reserved.