Journal of Physical Studies 6(3), 253–288 (2002) DOI: https://doi.org/10.30970/jps.06.253 THEORY OF SELF-SIMILAR STOCHASTIC SYSTEMS (PART I) A. I. Olemskoi, D. O. Kharchenko Sumy State University 2 Rimskiy-Korsakov Str., Sumy, UA–40007, Ukraine

The work is based on the use of the power-law asymptotics for the distribution function of self-similar systems. Starting from the obvious presentations we consider an anomalous nature of the particle walks, field representation of the stochastic process and time dependencies of most probable values and moments of stochastic variable. According to the equation for the distribution function a difference between main types of the particle walks (Brownian diffusion, Levy flights, subdiffusion) is explained. It is shown that the type of the motion is defined by the spatio-temporal distribution of intensity of the particle jump: if it takes an analytical form we have the ordinary diffusion in the case of nonanalytical dependence in the space we come to the Levy flights and nonanalitycal temporal dependence means the subdiffusion process. In the first case the distribution function acquires a power-law multiplier with respect to the coordinate, and power-law multiplier with respect to time in the second one. The relations between the dynamical exponent and indexes of the Levy flight and the subdiffusion as well as between fractal dimension and multiplicative noise exponent are defined. We show that the diffusion in the ultrametric space is defined with the help of the Fokker-Planck equation defined through the Jekson derivative. It is explained that the multiplicative noise breaks an additivity of the hierarchical system and mupltiplicativity of the probabilities. We find the general type of the fractional order motion equation, which in the limited cases are reduced to the wave equation, oscillating and transporting equation, Landau-Khalatnikov equation for conserved and nonconserved order parameter, Poisson equation and Debye's screening equation. In the nonlinear case it reduced to the Korteweg-de Wrize, sin-Gordon and a nonlinear Schrödinger equations. We show that the distribution function is defined by the exponential dependence with the exponent which is reduced to the standard action in the Euclidean field theory. For the white and coloured multiplicative noises the evolution of the more probable values of the stochastic variable and conjugate momentum is explored. We found that changing of the multiplicative noise exponent the system tests the reversible transition from the ordered state to disordered one and back. Form the Lagrangian and dissipative function, which are invariant with respect to the similarity transformations, the field equations are obtained.

PACS number(s): 02.50.-r, 47.53.+n, 72.70.+m

ps pdf