Limit theorems and governing equations for Lévy walks

M. Magdziarz, H.P. Scheffler, P. Straka and P. Zebrowski

Stochastic Processes and their Applications, 2015, vol. 125, issue 11, 4021-4038

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.

Keywords: Levy walk; Domain of attraction; Governing equation (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2015.05.014

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