Title Censored Lévy and related processes Title (croatian) Cenzurirani Lévyjevi i njima srodni procesi Author Vanja Wagner Mentor Zoran Vondraček (mentor)Committee member Miljenko Huzak (predsjednik povjerenstva)Committee member Zoran Vondraček (član povjerenstva)Committee member Nikola Sandrić (član povjerenstva)Granter University of Zagreb Defense date and country 2016-05-25, Croatia Scientific / art field, NATURAL SCIENCES Universal decimal classificationUDC ) 51 - Mathematics Abstract We examine three equivalent constructions of a censored rotationally symmetric Lévy process on an open set D - via the corresponding Dirichlet form, through the Feynman-Kac transform of the Lévy process killed outside of the set D and from the same killed process by the Ikeda-Nagasawa-Watanabe piecing together procedure. For a complete Bernstein function \(\phi\) satisfying condition (H): \[ a_1 \lambda^{\delta_1} \leq \frac{\phi(\lambda r)}{\phi(r)} \leq a_2 \lambda^{\delta_2}, \lambda \geq 1, r > 0 \] for some constants \(a_1, a_2 > 0\) and \(\delta_1, \delta_2 \in (0, 1)\), we prove the trace theorem for the Besov space of generalized smoothness \(H^{\phi (|\cdot|^2), 1}(\mathbb{R}^n)\) on \(n\)-sets. We analyze the behavior of the corresponding censored Brownian motion near the boundary \(\partial D\) and determine conditions under which the process approaches the boundary of the set D in finite time. Under a weaker condition (H1), i.e. (H) for \(\lambda, r \geq 1\), on the Laplace exponent \(\phi\) of the subordinator we prove the 3G inequality for Green functions of the subordinate Brownian motion on \(\kappa\)-fat open sets. Using this result we obtain the scale invariant Harnack inequality for the corresponding censored process. Finally, we consider a subordinate Brownian motion such that (H) holds and 0 is regular for itself. We establish a connection between this process and two related processes - censored process on the positive half-line and the absolute value of the subordinate Brownian motion killed at zero. We show that the corresponding Green functions on finite intervals away from 0 are comparable. Furthermore, we prove the Harnack inequality and the boundary Harnack principle for the absolute value of the subordinate Brownian motion killed at zero.
Abstract (croatian) Cenzurirani Lévyjev proces na otvorenom skupu D dobije se suzbijanjem skokova Lévyjevog procesa izvan skupa D restrikcijom pripadne Lévyjeve mjere na taj skup. U radu promatramo tri ekvivalentna pristupa u konstrukciji takvih procesa - preko pripadne Dirichletove forme, Feynman-Kacovom transformacijom Lévyjevog procesa ubijenog izvan skupa D te Ikeda-Nagasawa-Watanabe procedurom spajanja nezavisnih kopija Lévyjevog procesa ubijenog izvan skupa D. Dokazan je teorem o tragu na \(n\)-skupovima za generalizirane Besovljeve prostore \(H^{\psi,1}(\mathbb{R}^n)\) i to za karakteristične funkcije oblika \(\psi(x)=\phi(|x|^2), x \in \mathbb{R}^n\) gdje je \(\phi\) potpuna Bernsteinova funkcija koja zadovoljova svojstvo (H): \(a_1 \lambda^{\delta_1} \leq \frac{\phi(\lambda r)}{\phi(r)} \leq a_2 \lambda^{\delta_2}, \lambda \geq 1, r > 0\) za neke konstante \(a_1, a_2 > 0\) i \(\delta_1, \delta_2 \in (0, 1)\). Također, promatran je problem graničnog ponašanja cenzuriranog subordiniranog Brownovog gibanja s Laplaceovim eksponentom subordinatora \(\phi\), te su dani uvjeti pod kojima se proces približava rubu skupa D u konačnom vremenu. Uz pretpostavku da uvjet (H) vrijedi samo za \(\lambda, r \geq 1\) dokazana je 3G nejednakost za Greenovu funkciju tranzijentnog subordiniranog Brownovog gibanja na \(\kappa\)-debelim otvorenim skupovima. Korištenjem ovog rezultata pokazana je Harnackova nejednakost za pripadni cenzurirani proces. Promatramo subordinirano Brownovo gibanje za koje je 0 regularna točka za sebe te Laplaceov ekponent subordinatora zadovoljava uvjet (H). Uspostavlja se veza između ovog procesa i dva vezana procesa - cenzuriranog procesa na \((0, \infty)\) i apsolutne vrijednosti pripadnog procesa ubijenog u nuli. Pokazano je da su pripadne Greenove funkcije procesa ubijenih izvan konačnog intervala \((a, b)\), za \(0 < a < b\), usporedive. Nadalje, dokazana je Harnackova nejednakost i granični Harnackov princip za apsolutnu vrijednost subordiniranog Brownovog gibanja ubijenog u 0.
Keywords
Levy process
censored process
Harnack inequality
boundary Harnack inequality
Dirichlet form
trace theorem
Keywords (croatian)
Lévyjev proces
cenzurirani proces
Harnackova nejednakost
granični Harnackov princip
Dirichletova forma
teorem o tragu
Language english URN:NBN urn:nbn:hr:217:952135 Study programme Title: Mathematics Type of resource Text Extent ii, 108 str. File origin Born digital Access conditions Open access Terms of use Created on 2019-03-22 12:20:16
