Theory of stochastic systems with singular multiplicative noise

Noisy, interacting, stochastic systems are analyzed for the case in which their noise intensity varies with the hydrodynamic mode amplitude x according to the power law x^2a, x &z.ele; [0, 1]. It is shown that the phase space domain of definition of the stochastic variable x forms a self-affine set of fractal dimensionality D = 2(1-a). Using the gauge procedure, a system of calculus is chosen which is not reducible either to the Ito case or the Stratonovich case. By generalizing the microscopic picture of phase transitions it is demonstrated that the system may reduce its symmetry (for 1 < D ≤ 2) or lose ergodicity (for 0 < D ≤ 1). Over the entire interval D &z.ele; [0, 2], a noise-induced transition is shown to be possible.