Banca de DEFESA: DIEGO ALEJANDRO MONROY ALVAREZ

Uma banca de DEFESA de MESTRADO foi cadastrada pelo programa.
STUDENT : DIEGO ALEJANDRO MONROY ALVAREZ
DATE: 13/06/2023
TIME: 14:00
LOCAL: Através de Videoconferência: https://meet.google.com/azy-rhmr-xjz
TITLE:

Jacobi Polynomials Approach to the Random Search Problem in One Dimension

KEY WORDS:

Random searches. Lévy 𝛼-stable distribution. Classical Jacobi polynomials.

PAGES: 90
BIG AREA: Ciências Exatas e da Terra
AREA: Física
SUMMARY:

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.

COMMITTEE MEMBERS:
Presidente - 2287840 - ERNESTO CARNEIRO PESSOA RAPOSO
Externo à Instituição - MARCOS GOMES ELEUTERIO DA LUZ - UFPR
Interno - 1129671 - MAURICIO DOMINGUES COUTINHO FILHO