Jacobi polynomials approach to the random search problem in one dimension

Abstract

Lévy processes, either flights or walks, have attracted a great deal of attention from diverse fields. They have been successfully applied to model anomalous transport phenomena in superconductors, turbulence, sunlight scattering in clouds, spectroscopy and random lasers. In ecology, there are numerous evidence that living organism often forage "non-gaussianly", a behaviour that, in theory, results in more efficient searches. Short-term deviations from normality have also been observed in financial assets prices and Lévy processes have been applied to analyse market microstructure and market friction. We address the problem of one-dimensional symmetric Lévy flights that take place in a finite interval with absorbing endpoints, i.e. the target sites. Pure Lévy flights are by no means easy to tackle analitically, hence the jump step length is sampled from a power-law (Pareto I) distribution with shape parameter 0 < α < 2 thus resembling the asymptotic heavy-tailed behaviour of the Lévy α-stable distribution. For such simplified system, closed-form expressions have been reported in the literature for the absorption probability at a specific target, the mean number of steps and the mean path length before a target is encountered, of which the last two quantities are of special interest since they are related to the mean first-passage time of Lévy flyers and walkers respectively. Those approximate closed-form expressions have been obtained by means of inversion formulae related to fractional integro-differential equations and perform reasonably well provided that the departure site is not too close to the targets and away from the Gaussian regime. This work not only intends to revisit the aforementioned approach but also to explore alternative methods, such as the spectral relationship method using classical Jacobi polynomials. This method allows the inclusion of correction terms that are difficult to handle with inversion formulae. The obtained solutions predict the simulated results more accurately and in broader ranges of the stability index and the departure site location than their inversion formulae counterparts. As a drawback, one must resort to numerical methods and regularization techniques to deal with the instability arising for the ill-conditioned nature of problem.