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Lévy, Ornstein–Uhlenbeck, and Subordination: Spectral vs. Jump Description

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Abstract

Unlike Brownian motion, which propagates diffusively and whose sample-path trajectories are continuous, non-Brownian Lévy motions propagate via jumps (flights) and their sample-path trajectories are purely discontinuous. When analyzing systems involving non-Brownian Lévy motions, the common practice is to use either spectral or fractional-calculus methods. In this manuscript we suggest an alternative analytical approach: using the Poisson-superposition jump structure of non-Brownian Lévy motions. We demonstrate this approach in two exemplary topics: (i) systems governed by L évy-driven Ornstein–Uhlenbeck dynamics; and, (ii) systems subject to temporal Lévy subordination. We show that this approach yields answers and insights that are not attainable using spectral methods alone.

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Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel

    Iddo Eliazar

  2. School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Joseph Klafter

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  1. Iddo Eliazar
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  2. Joseph Klafter
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Correspondence to Iddo Eliazar.

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Eliazar, I., Klafter, J. Lévy, Ornstein–Uhlenbeck, and Subordination: Spectral vs. Jump Description. J Stat Phys 119, 165–196 (2005). https://doi.org/10.1007/s10955-004-2710-9

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