Limit theorems and governing equations for Lévy walks

Abstract

The Lévy Walk is the process with continuous sample paths which arises from consecutive linear motions of i.i.d. lengths with i.i.d. directions. Assuming speed 1 and motions in the domain of β-stable attraction, we prove functional limit theorems and derive governing pseudo-differential equations for the law of the walker’s position. Both Lévy Walk and its limit process are continuous and ballistic in the case β∈(0,1). In the case β∈(1,2), the scaling limit of the process is β-stable and hence discontinuous. This result is surprising, because the scaling exponent 1/β on the process level is seemingly unrelated to the scaling exponent 3−β of the second moment. For β=2, the scaling limit is Brownian motion.