Exact propagator for generalized Ornstein-Uhlenbeck processes.
A closed form expression for the propagator is derived, in terms of modified Bessel functions, for the Fokker-Planck equation for a physically important generalization of the Ornstein-Uhlenbeck process where the diffusion constant D(p) is a function of the momentum. The closed form is found for the general case D(p) approximately |p|(-alpha) where alpha > or =0 and leads to the standard Gaussian form for alpha=0 . The propagator for the specific case D(p) approximately |p|(-1) is used to derive analytic expressions for probability distributions and correlation coefficients. An exact expression is found for the constant of proportionality for the anomalous diffusion of the mean-square displacement of a particle at short times.
Mota-Furtado
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Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom. f.motefurtado@rhul.ac.uk
O'Mahony
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PF