Naslov
Censored Levy and Related Processes

Autori
Wagner, Vanja

Vrsta, podvrsta i kategorija rada
Ocjenski radovi, doktorska disertacija

Fakultet
Prirodoslovno-matematički fakultet - Matematički odsjek

Mjesto
Zagreb

Datum
25.05

Godina
2016

Stranica
103

Mentor
Vondraček, Zoran

Ključne riječi
Levy process ; censored process ; Harnack inequality ; boundary Harnack inequality ; Dirichlet form ; trace theorem

Sažetak
We examine three equivalent constructions of a censored rotationally symmetric Leve process on an open set $D$ - via the corresponding Dirichlet form, through the Feynman-Kac transform of the Levy 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): $$ a_1 \lambda^{; ; ; ; \delta_1}; ; ; ; \le \frac{; ; ; ; \phi(\lambda r)}; ; ; ; {; ; ; ; \phi(r)}; ; ; ; \le a_2\lambda^{; ; ; ; \delta_2}; ; ; ; , lambda\ge 1, r>0, $$ 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}; ; ; ; on $n$-sets. We analyze the behavior of the corresponding censored subordinate Brownian motion near the boundary $\partial D$ and determine conditions under which the process approaches the boundary of the set $D$ in nite time. Under a weaker condition (H1), i.e. (H) for $\lambda, r\ge 1$, 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 nite 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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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