Ornstein–Uhlenbeck–Cauchy process

2000/10/01 by Piotr Garbaczewski, Robert Olkiewicz · 3 citations
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Statistical Mechanics and Entropy #stochastic dynamics and bifurcation

paper · doi:10.1063/1.1290054

Abstract

We combine earlier investigations of linear systems subject to Lévy fluctuations with recent attempts to give meaning to so-called Lévy flights in external force fields. We give a complete construction of the Ornstein–Uhlenbeck–Cauchy process as a fully computable paradigm example of Doob’s stable noise-supported Ornstein–Uhlenbeck process. Despite the nonexistence of all moments, we determine local characteristics (forward drift) of the process, generators of forward and backward dynamics, and relevant (pseudodifferential) evolution equations. The induced nonstationary spatial process is proved to be Markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense.

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