Wagner, Vanja (2016) Censored Lévy and related processes. Doctoral thesis, Faculty of Science > Department of Mathematics.

PDF
Language: English Download (809kB)  Preview 
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 FeynmanKac transform of the Lévy process killed outside of the set D and from the same killed process by the IkedaNagasawaWatanabe 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 halfline 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.
Item Type:  Thesis (Doctoral thesis) 

Supervisor:  Vondraček, Zoran 
Date:  2016 
Number of Pages:  108 
Subjects:  NATURAL SCIENCES > Mathematics NATURAL SCIENCES > Mathematics > Probability Theory and Statistics 
Divisions:  Faculty of Science > Department of Mathematics 
Depositing User:  Iva Prah 
Date Deposited:  01 Jun 2016 11:27 
Last Modified:  01 Jun 2016 11:27 
URI:  http://digre.pmf.unizg.hr/id/eprint/4881 
Actions (login required)
View Item 