Zoran Vondraček

Bernstein Functions: Theory and Applications

This text is a self-contained and unified approach to Bernstein functions and their subclasses, bringing together old and establishing new connections. Applications of Bernstein functions in different fields of mathematics are given, with special attention to interpretations in probability theory. An extensive list of complete Bernstein functions with their representations is provided. A self-contained and unified approach to the topic With applications to various fields of mathematics, such as probability theory, potential theory, operator theory, integral equations, functional calculi and complex analysis With an extensive list of complete Bernstein functions.

Ruin probabilities and decompositions for general perturbed risk processes

We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, and the perturba- tion being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival proba- bility of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.

Potential Theory of Subordinate Brownian Motions Revisited

The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity. This extends some results previously obtained under stronger conditions.

Two-sided Green function estimates for killed subordinate Brownian motions

A subordinate Brownian motion is a Levy process that can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is �φ(�Δ), where φ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with φ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of independent relativistic stable processes and much more. We give sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded κ-fat open set D.W henD is a bounded C 1,1 open set, we establish an explicit form of the estimates in terms of the distance to the boundary. As a consequence of such sharp Green function estimates, we obtain a boundary Harnack principle in C 1,1 open sets with explicit rate of decay.

Potential Theory of Subordinate Brownian Motion

Classical and boundary potential theory is presented for the subordinators and the subordinate processes. Main examples of subordinate processes include stable processes, relativistic stable processes, and geometrically stable processes.

Uniform boundary Harnack principle for rotationally symmetric Lévy processes in general open sets

In this paper we prove the uniform boundary Harnack principle in general open sets for harmonic functions with respect to a large class of rotationally symmetric purely discontinuous Lévy processes.

Boundary Harnack principle for Δ+Δ^{/2}

For d � 1 and � 2 (0,2), consider the family of pseudo differential operators f �+b� �/2 ;b 2 (0,1)g on R d that evolves continuously fromto � + � �/2 . In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to �+b� �/2 (or equivalently, the sum of a Brownian motion and an independent symmetric�-stable process with constant multipleb 1/� ) inC 1,1 open sets. Here a uniform BHP means that the comparing constant in the BHP is independent of b 2 (0,1). Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to � +b� �/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.

On suprema of Lévy processes and application in risk theory

Let

Some remarks on special subordinators

wh{; ; ; X}; ; ; =C-Y

Asymptotics of First-Passage Time Over a One-Sided Stochastic Boundary

where