Density in small time for Lévy processes

Abstract

The density of real-valued Lévy processes is studied in small time under the assumption that the process has many small jumps. We prove that the real line can be divided into three subsets on which the density is smaller and smaller: the set of points that the process can reach with a finite number of jumps (Δ-accessible points); the set of points that the process can reach withan infinite number of jumps (asymptotically Δ-accessible points); and the set of points that the process cannot reach by jumping (Δ-inaccessible points).