Limit Theorems and Governing Equations for Levy Walks

TL;DR

This paper establishes limit theorems and governing equations for Levy Walks, revealing different behaviors depending on the stability parameter beta, including continuous, discontinuous, and Brownian motion limits.

Contribution

It provides the first rigorous derivation of functional limit theorems and pseudo-differential equations for Levy Walks under beta-stable attraction.

Findings

For beta in (0,1), Levy Walks are continuous and ballistic.

For beta in (1,2), the limit process is beta-stable and discontinuous.

For beta=2, the limit is Brownian motion.

Abstract

The Levy 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 beta-stable attraction, we prove functional limit theorems and derive governing pseudo-differential equations for the law of the walker's position. Both Levy Walk and its limit process are continuous and ballistic in the case beta in (0,1). In the case beta in (1,2), the scaling limit of the process is beta-stable and hence discontinuous. This case exhibits an interesting situation in which scaling exponent 1/beta on the process level is seemingly unrelated to the scaling exponent 3-beta of the second moment. For beta = 2, the scaling limit is Brownian motion.