Multiplicative stochastic differential equations with noise-induced transitions

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Abstract
The author investigates a class of linear multiplicative stochastic differential equations and demonstrates the existence of a striking noise-induced transition in the structure of the resulting asymptotic stationary probability distribution for the dependent variable. The transition amounts to a change from a bounded distribution to an unbounded one with only a finite number of convergent moments. It occurs when the range of fluctuation of one of the variables driven by the underlying stochastic process increases sufficiently to permit changes of sign for the variable. It seems likely that the phenomenon is a general one and occurs in a wider class of models than that discussed in this paper. He obtains explicit results for simple cases which he confirms by appropriate numerical simulations. This gives him the opportunity of assessing the applicability of perturbation theory which is one of the few calculational methods employed on these models up until now.

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