L2 Diffusion Approximation for Slow Motion in Averaging

Assuming that the fast motion in averaging is sufficiently well mixing we show that the slow motion can be approximated in the L2-sense by a diffusion solving Hasselmann's nonlinear stochastic differential equation and which provides a much better approximation than the one suggested by the averaging principle. Previously, only weak limit theorems in averaging were known which cannot justify, in principle, a nonlinear diffusion approximation of the slow motion.

[1]  W. Philipp,et al.  Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales , 1991 .

[2]  M. Freidlin,et al.  Functional Integration and Partial Differential Equations. (AM-109), Volume 109 , 1985 .

[3]  R. Has’minskiĭ.  On Stochastic Processes Defined by Differential Equations with a Small Parameter , 1966 .

[4]  Walter Philipp,et al.  Approximation by Brownian motion for Gibbs measures and flows under a function , 1984, Ergodic Theory and Dynamical Systems.

[5]  Walter Philipp,et al.  Almost Sure Invariance Principles for Partial Sums of Mixing $B$-Valued Random Variables , 1980 .

[6]  W. Philipp,et al.  The Problem of Embedding Vector-Valued Martingales in a Gaussian Process , 1991 .

[7]  A. Borodin,et al.  A limit theorem for solutions of di erential equations with random right-hand side , 1978 .

[8]  Y. Kifer.  Limit theorems in averaging for dynamical systems , 1995, Ergodic Theory and Dynamical Systems.

[9]  W. Philipp,et al.  Approximation Thorems for Independent and Weakly Dependent Random Vectors , 1979 .

[10]  V. P. Leonov,et al.  On the Dispersion of Time-Dependent Means of a Stationary Stochastic Process , 1961 .

[11]  Y. Kifer.  The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point , 1981 .

[12]  Yuri Kifer,et al.  Averaging in dynamical systems and large deviations , 1992 .

[13]  G. Belitskii,et al.  EQUIVALENCE AND NORMAL FORMS OF GERMS OF SMOOTH MAPPINGS , 1978 .

[14]  R. Khas'minskii.  A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides , 1966 .