# Censored Lévy and related processes

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

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Language: English

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.