On the averaging theorems for stochastic perturbation of conservative linear systems

TL;DR

The work addresses stochastic perturbations of linear conservative systems with purely imaginary spectra, focusing on the long-time behavior of action variables via slow-fast action-angle dynamics and Krylov–Bogoliubov averaging. It constructs an effective equation by angle-averaging the drift and diffusion, yielding a drift $F(I)$ and diffusion $K(I)$ for the action vector $I$, derived from the averaged drift and the matrix $B(a)$ of the diffusion. Under the assumption of uniform ellipticity of the diffusion, the authors show that the limiting action dynamics is independent of the Hamiltonian component of the perturbation, and they extend the result to a modified effective equation through a rigorous δ-cutoff and martingale analysis. The results provide a rigorous stochastic averaging framework for non-resonant linear systems under small random perturbations and underpin future averaging analyses in stochastic partial differential equations.

Abstract

For stochastic perturbations of linear systems with non-zero pure imaginary spectrum we discuss the averaging theorems in terms of the slow-fast action-angle variables and in the sense of Krylov-Bogoliubov. Then we show that if the diffusion matrix of the perturbation is uniformly elliptic, then in all cases the averaged dynamics does not depend on a hamiltonian part of the perturbation.

Theorems & Definitions (17)

• Example 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4
• Proposition 2.5
• Corollary 2.6
• Proposition 2.7
• Theorem 3.1
• proof
• Lemma 3.2