Subordinate random walks

doctoral thesis

• Title (croatian): Subordinirane slučajne šetnje
• Author: Stjepan Šebek
• Mentor: Zoran Vondraček (mentor)
• Committee member: Bojan Basrak (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: 2019-05-16, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

In this thesis, we consider a large class of subordinate random walks on the integer lattice \(\mathbb{Z}^d\) via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions at zero. Subordination is a procedure for obtaining new process from the original one. The new process may differ very much from the original process, but the properties of this new process can be understood in terms of the original process. Main results of the thesis are the elliptic Harnack inequality and n-step transition probability estimates for subordinate random walks. In order to obtain the elliptic Harnack inequality, we first establish estimates for one-step transition probabilities, the Green function and the Green function of a ball. The main tools we apply to get \(n\)-step transition probability estimates for subordinate random walks are the parabolic Harnack inequality and appropriate bounds for the transition kernel of the corresponding continuous time random walk.

Abstract (croatian)

U ovoj disertaciji promatramo veliku klasu subordiniranih slučajnih šetnji na cjelobrojnoj mreži \(\mathbb{Z}^d\) dobivenih pomoću subordinatora s Laplaceovim eksponentima koji su potpune Bernsteinove funkcije koje zadovoljavaju neke blage uvjete skaliranja u nuli. Subordinacija je procedura dobivanja novog procesa na temelju originalnog procesa. Iako se novi proces može dosta razlikovati od originalnog, svojstva dobivenog procesa mogu se shvatiti u terminima originalnog procesa. Glavni rezultati do kojih dolazimo su eliptička Harnackova nejednakost te ocjene na prijelazne vjerojatnosti za subordinirane slučajne šetnje. Kako bismo dobili eliptičku Harnackovu nejednakost, prvo dokazujemo ocjene za jednokoračne prijelazne vjerojatnosti, Greenovu funkciju te Greenovu funkciju kugle. Glavne tehnike koje koristimo kako bismo dobili ocjene za \(n\)-koračne prijelazne vjerojatnosti za subordinirane slučajne šetnje su parabolička Harnackova nejednakost i odgovarajuće ocjene za prijelaznu jezgru pripadajuće neprekidno vremenske slučajne šetnje.

• Keywords: Random walk; Subordination; Harnack inequality; Green function
• Keywords (english): slučajna šetnja; subordinacija; Harnackova nejednakost; Greenova funkcija
• Language: english
• URN:NBN: urn:nbn:hr:217:924439
• 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, 86 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-05-28 11:21:28