Generalization of the Blumenthal–Getoor index to the class of homogeneous diffusions with jumps and some applications

Abstract

We introduce the probabilistic symbol for the class of homogeneous diffusions with jumps (in the sense of Jacod/Shiryaev). This concept generalizes the well-known characteristic exponent of a L\'{e}vy process. Using the symbol, we introduce eight indices which generalize the Blumenthal-Getoor index $\beta$ and the Pruitt index $\delta$. These indices are used afterwards to obtain growth and H\"{o}lder conditions of the process. In the future, the technical main results will be used to derive further fine properties. Since virtually all examples of homogeneous diffusions in the literature are Markovian, we construct a process which does not have this property.