Periodic homogenization for Levy-type processes

doctoral thesis

• Title (croatian): Periodička homogenizacija za procese Levyjevog tipa
• Author: Ivana Valentić
• Mentor: Nikola Sandrić (mentor)
• Committee member: Vanja Wagner (predsjednik povjerenstva)
• Committee member: Nikola Sandrić (član povjerenstva)
• Committee member: Kristina Ana Škreb (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

The main goal of this thesis is to discuss periodic homogenization of a Lévy-type pseudodifferential operator. Our approach to this problem is based on probabilistic techniques. More precisely, as the main result we show that the appropriately centered and scaled Lévy-type process (LTP) generated by this operator converges weakly to a Brownian motion with covariance matrix given in terms of the operator coefficients. We specially focus on a class of Lévy-type processes admitting “small jumps” only and a class of diffusion processes having degenerate diffusion term. These results generalize and refine the classical and well-known results related to periodic homogenization of diffusion process and of Lévy-type process in balanced form. In order to resolve these problems, it is necessary to combine both probabilistic and analytical approaches and tools, such as theory of semimartingales, stochastic stability theory and theory of integro-differential equations.

Abstract (croatian)

Glavni cilj ove disertacije je diskutirati periodičku homogenizaciju pseudo-diferencijalnog operatora Lévyjevog tipa. Naš pristup ovom problemu bazira se na vjerojatnosnim metodama. Preciznije, kao glavni rezultat dokazujemo da odgovarajuće centriran i skaliran proces Lévyjevog tipa generiran takvim operatorom slabo konvergira prema Brownovom gibanju s kovarijacijskom matricom danom u terminima koeficijenata operatora. Posebno se koncentriramo na klasu procesa Lévyjevog tipa koji dozvoljavaju samo “male skokove” i na klasu procesa difuzija s degeneriranim difuzijskim koeficijentom. Ti rezultati generaliziraju i produbljuju klasične i dobro poznate rezultate vezane uz periodičku homogenizaciju difuzije i procesa Lévyjevog tipa u balansiranom obliku. Kako bismo razriješili ove probleme nužno je kombinirati vjerojatnosni i analitički pristup i metode, kao što su teorija semimartingala, teorija stohastičke stabilnosti i teorija integro-diferencijalnih jednadžbi.

• Keywords: Brownian motion; Markov processes; Lévy-type processes; Feller processes; semimartingales; ergodicity; central limit theorem; homogenization; stochastic differential equations; partial differential equations; Itô’s formula; Feynman-Kac formula; Poisson equation
• Keywords (croatian): Brownovo gibanje; Markovljevi procesi; Procesi Lévyjevog tipa; Fellerovi procesi; semimartingali; ergodičnost; centralni granični teorem; homogenizacija; stohastičke diferencijalne jednadžbe; parcijalne diferencijalne jednadžbe; Itôva formula; Poissonova jednadzba; Feynman-Kacova formula
• Language: english
• URN:NBN: urn:nbn:hr:217:462190
• Promotion: 2020
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: v, 121 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-12-04 12:57:48