Potential Theory of Subordinate Brownian Motion

Abstract

The materials covered in the second part of the book are based on several recent papers, primarily [132], [139], [148] and [146]. The main effort here was given to unify the exposition of those results, and in doing so we also eradicated the typos in these papers. Some new materials and generalizations are also included. Here is the outline of Chapter 5. In Section 5.2 we recall some basic facts about subordinators and give a list of examples that will be useful later on. This list contains stable subordinators, relativistic stable subordinators, subordinators which are sums of stable subordinators and a drift, gamma subordinators, geometric stable subordinators, iterated geometric stable subordinators and Bessel subordinators. All of these subordinators belong to the class of special subordinators (even complete Bernstein subordinators). Special subordinators are important to our approach because they are precisely the ones whose potential measure restricted to (0,?) has a decreasing density u. In fact, for all of the listed subordinators the potential measure has a decreasing density u. In the last part of the section we study asymptotic behaviors of the potential density u and the Lévy density of subordinators by use of Karamata’s and de Haan’s Tauberian and monotone density theorems.