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Limit theorems and governing equations for Lévy walks

Author

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

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.

Suggested Citation

  • Magdziarz, M. & Scheffler, H.P. & Straka, P. & Zebrowski, P., 2015. "Limit theorems and governing equations for Lévy walks," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4021-4038.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:11:p:4021-4038
    DOI: 10.1016/j.spa.2015.05.014
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    References listed on IDEAS

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    Cited by:

    1. Peggy Cénac & Arnaud Ny & Basile Loynes & Yoann Offret, 2019. "Persistent Random Walks. II. Functional Scaling Limits," Journal of Theoretical Probability, Springer, vol. 32(2), pages 633-658, June.

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