Limit theorems and governing equations for Levy walks

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 epsilon (0, 1). In the case beta epsilon (1, 2), the scaling limit of the process is beta-stable and hence discontinuous. This result is surprising, because the scaling exponent 1/beta on the process level is seemingly unrelated to the scaling exponent 3 - beta of the second moment. For = 2, the scaling limit is Brownian motion. (C) 2015 Elsevier B.V. All rights reserved.