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 \in [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 \leqslant 2) or lose ergodicity (for 0 < D \leqslant 1). Over the entire interval D \in [0, 2], a noise-induced transition is shown to be possible.

PACS: 05.40.+j, 05.70.Fh, 64.60.−i, 82.20.Fd (all) DOI: 10.1070/PU1998v041n03ABEH000377 URL: https://ufn.ru/en/articles/1998/3/c/ 000073306100003 Citation: Olemskoi A I "Theory of stochastic systems with singular multiplicative noise" Phys. Usp. 41 269–301 (1998) BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks Оригинал: Олемской А И «Теория стохастических систем с сингулярным мультипликативным шумом» УФН 168 287–321 (1998); DOI: 10.3367/UFNr.0168.199803c.0287