The theory of Brownian motion as described by nonlinear Langevin equations and the corresponding Fokker-Planck equations is discussed. The general problems of the theory of nonlinear Brownian motion considered are: Brownian motion in a medium with nonlinear friction; the critical analysis of three forms of the relevant Langevin and Fokker-Planck equations (Ito’s form, Stratonovich’s form, and the kinetic form); the Smoluchowski equations and master equations for different cases; two methods of transition from master equation to Fokker-Planck equation; master equations for one-step processes; traditional and nontraditional definition of transition probabilities; evolution of free energy and entropy in Brownian motion; Lyapunov functionals. The following particular examples are considered: Brownian motion in self-oscillatory systems; H-theorem for the van der Pol oscillator; S-theorem; oscillator with inertial nonlinearity; bifurcation of energy of the limiting cycle; oscillator with multistable stationary states; oscillators in discrete time; bifurcations of energy of the limiting cycle and the period of oscillations; criterion of instability upon transition to discrete time, based on the H-theorem; Brownian motion of quantum atoms oscillators in the equilibrium electromagnetic field; Brownian motion in chemically reacting systems; partially ionised plasmas; the Malthus-Verhulst process.

PACS: 05.40.+j DOI: 10.1070/PU1994v037n08ABEH000038 URL: https://ufn.ru/en/articles/1994/8/b/ A1994PJ71400002 Citation: Klimontovich Yu L "Nonlinear Brownian motion" Phys. Usp. 37 737–766 (1994) BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks Оригинал: Климонтович Ю Л «Нелинейное броуновское движение» УФН 164 811–844 (1994); DOI: 10.3367/UFNr.0164.199408b.0811