Multiplicative stochastic processes in statistical physics

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Abstract
For a large class of nonlinear stochastic processes with pure multiplicative fluctuations the corresponding time-dependent Fokker-Planck-equation is solved exactly by analytic methods. A universal eigenvalue spectrum and the corresponding set of eigenfunctions are obtained in closed form. The eigenvalue spectrum consists of a discrete as well as a continuous part. To emphasize the significance of the model proposed for the description of more-general stochastic processes the authors investigate its stability with respect to the inclusion of weak additive fluctuations. A discussion of the differences in the static as well as the dynamic behavior of multiplicative and additive stochastic processes is given in detail. It is shown explicitly how internal as well as externally imposed fluctuations can lead to multiplicative stochastic processes. The applications of the results to various fields such as nonlinear optics—subharmonic generation, parametric three-wave mixing, Raman scattering—electronic devices, autocatalytic chemical reactions, and population dynamics are given. In particular, a comparison with recent experiments by S. Kabashima et al., who investigated the statistical properties of electronic parametric oscillators driven by external noise, is carried out.