Carleman imbedding of multiplicative stochastic processes

Abstract

A class of exactly solvable nonlinear stochastic models with multiplicative Gaussian white noise is here solved by a method of linear imbedding. The models describe a continuous instability with fluctuations of the bifurcation parameter. The method of solution is a generalization of an idea, originally introduced by Carleman for the solution of deterministic rate equations. Our application of this method to the stochastic models studied here provides further insight into the applicability of the method. The solutions we find are compared with results, which have been obtained earlier by Fokker-Planck methods. Complete agreement with the Fokker-Planck results is found, contrary to recent claims by Brenig and Banai and also in disagreement with recent results of Suzuki et al. The method presented here also allows us, for the first time, to obtain solutions in a domain of parameter space where the Fokker-Planck equation has not been solved as yet.