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)Medline RefWorks RT Journal T1 Theory of stochastic systems with singular multiplicative noise A1 Olemskoi,A.I. PB Physics-Uspekhi PY 1998 FD 10 Mar, 1998 JF Physics-Uspekhi JO Phys. Usp. VO 41 IS 3 SP 269-301 DO 10.1070/PU1998v041n03ABEH000377 LK https://ufn.ru/en/articles/1998/3/c/ Оригинал: Олемской А И «Теория стохастических систем с сингулярным мультипликативным шумом» УФН 168 287–321 (1998); DOI: 10.3367/UFNr.0168.199803c.0287