Diffusive random process approximation in certain nonstationary statistical problems of physics

The review considers, on the basis of a unified approach, the problem of Brownian motion in nonlinear dynamic systems, including a linear oscillator acted upon by random forces, parametric resonance in an oscillating system with random parameters, turbulent diffusion of particles in a random-velocity field, and diffusion of rays in a medium with random inhomogeneities of the refractive index. The same method is used to consider also more complicated problems such as equilibrium hydrodynamic fluctuations in an ideal gas, description of hydrodynamic turbulence by the method of random forces, and propagation of light in a medium with random inhomogeneities. The method used to treat these problems consists of constructing equations for the probability density of the system or for its statistical moments, using as the small parameter the ratio of the characteristic time of the random actions to the time constant of the system (in many problems, the role of the time is played by one of the spatial coordinates). The first-order approximation of the method is equivalent to replacement of the real correlation function of the action by a $\delta$ function; this yields equations for the characteristics in closed form. The method makes it possible to determine also higher approximations in terms of the aforementioned first-order small parameter.