Article

Journal of Statistical Physics

, Volume 119, Issue 1, pp 165-196

First online:

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

  • Iddo EliazarAffiliated withDepartment of Mathematics, Bar-Ilan University Email author 
  • , Joseph KlafterAffiliated withSchool of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University

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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.

Keywords

Non-Brownian Lévy motions selfsimilar Lévy motions Poisson superposition Lévy-driven Ornstein–Uhlenbeck dynamics temporal Lévy subordination