Generalized Ornstein-Uhlenbeck processes

Bezuglyy, V.; Mehlig, B.; Wilkinson, Michael; Nakamura, K. and Arvedson, E. (2006). Generalized Ornstein-Uhlenbeck processes. Journal of Mathematical Physics, 47(7)

DOI: https://doi.org/10.1063/1.2206878

Abstract

We solve a physically significant extension of a classic problem in the theory of diffusion, namely the Ornstein-Uhlenbeck process [Ornstein and Uhlenbeck, Phys. Rev. 36, 823 (1930)]. Our generalized Ornstein-Uhlenbeck systems include a force which depends upon the position of the particle, as well as upon time. They exhibit anomalous diffusion at short times, and non-Maxwellian velocity distributions in equilibrium. Two approaches are used. Some statistics are obtained from a closed-form expression for the propagator of the Fokker-Planck equation for the case where the particle is initially at rest. In the general case we use spectral decomposition of a Fokker-Planck equation, employing nonlinear creation and annihilation operators to generate the spectrum which consists of two staggered ladders. (c) 2006 American Institute of Physics.

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