Moments of the distributio function and kinetic equation for stochastic motion of a nonlinear oscillator

An analytic approach to study of the stochastic motion of a nonlinear system in a periodic external potential is developed. In contrast to a number of other approaches, no additional external random parameters are introduced a priori. A method for calculating the moments of the distribution function is constructed. In particular, the problem of calculating the diffusion coefficient is reduced to the solution of an infinite inhomogeneous system of linear equations. In the limit of large values of Chirikov's stochasticity parameter $K$, this system simplifies strongly and reduces to a system of two equations up to terms of order $1/\sqrt{4K}$. In this limit, the diffusion coefficient can be readily found in explicit form. In the leading approximation in the parameter $1/\sqrt{4K}$ a closed expression is obtained for the generating function of the moments of the distribution function. It differs strongly from the standard Gauss[an expression. A kinetic equation is obtained for the coarse-grained distribution function. Although it differs from the standard diffusion equation that is generally used, its solution tends asymptotically at large times to the Gaussian distribution.