Ascione, Giacomo (2021) Probabilistic approach to non-local equations. [Tesi di dottorato]


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Item Type: Tesi di dottorato
Lingua: English
Title: Probabilistic approach to non-local equations
Date: 8 February 2021
Number of Pages: 180
Institution: Università degli Studi di Napoli Federico II
Department: Matematica e Applicazioni "Renato Caccioppoli"
Dottorato: Matematica e applicazioni
Ciclo di dottorato: 33
Coordinatore del Corso di dottorato:
Mantegazza, Carlo MariaUNSPECIFIED
Date: 8 February 2021
Number of Pages: 180
Uncontrolled Keywords: Non-local operator; Subordinator; Bernstein function
Settori scientifico-disciplinari del MIUR: Area 01 - Scienze matematiche e informatiche > MAT/05 - Analisi matematica
Area 01 - Scienze matematiche e informatiche > MAT/06 - Probabilità e statistica matematica
Date Deposited: 18 Feb 2021 18:21
Last Modified: 07 Jun 2023 10:27


The work focuses on probabilistic representation of solutions of non-local equations, where the considered non-local operators are either Caputo-type derivatives or trasformations of the Laplace operator via Bernstein functions. The first chapter is devoted to the introduction of the main tools, that is to say Bernstein functions, subordinators and their inverse processes, Caputo-type derivatives and Bochner subordination. In the second chapter we focus on theoretical results concerning probabilistic representation of non-local equations involving Caputo-type derivatives. First we study a general theory for abstract Cauchy problems involving such kind of derivatives, focusing also on the eigenfunctions of such Caputo-type derivatives and Gronwall-type inequalities. Then we exploit the link between time-changed birth-death processes and abstract Cauchy problems in suitable Banach sequence spaces, via a spectral decomposition approach. Then we consider how the Fokker-Planck equation of a non-Markov Gaussian process changes after applying a time-change via the inverse of a subordinator. Finally, we consider exit times of time-changed processes, their asymptotic problems and the link between their survival probability and solutions of non-local (in time) parabolic PDEs. The third chapter is devoted to applications of the previously presented theoretical results. In particular we focus on queueing theory and computational neuroscience. Moreover, we also exploit some simulation properties to work with such processes. In the fourth and last chapter we focus on non-local operators in space with two exemplary problems. The first one concerns the integral representation of Bernstein functions of the Laplace operator. In particular we prove asymptotic properties of the singular kernel of such integral representations depending on asymptotic properties of the Levy measure of the considered Bernstein function. The second problem deals with spectral properties of a Marchaud-type operator on the sphere. In particular, we prove an identity involving the first eigenvalue of the aforementioned operator and moments of the length of random segments in the unit ball.


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