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Two approaches to average stochastic perturbations of integrable systems

Sergei Kuksin

TL;DR

This work analyzes the long-time behavior of small stochastic perturbations of integrable Hamiltonian systems in two complementary ways: (i) a fast-slow (action-angle) approach yielding an averaged equation for the actions, and (ii) an effective equation obtained by Krylov--Bogolyubov-type averaging in the original variables. The actions converge in distribution to the solution of the averaged dynamics under broad conditions, while an auxiliary effective equation provides a unique, globally well-behaved limit that captures the leading behavior of the actions for all times; local versions and explicit applications (chains of oscillators and damped/driven systems) illustrate the framework. The results highlight the stabilizing role of noise and friction, show when averaging yields unique limits, and contrast stochastic with deterministic perturbations, with extensions to local normal forms and potential ties to SPDEs. Overall, the paper offers a rigorous synthesis of averaging techniques for stochastic perturbations of integrable systems and clarifies when the two main approaches yield robust, time-uniform descriptions of action dynamics.

Abstract

We discuss two approaches to study the long-time behaviour and infinite-time behaviour of solutions for integrable hamiltonian systems under small stochastic perturbations. Then we compare these results with those for deterministic perturbations of integrable systems.

Two approaches to average stochastic perturbations of integrable systems

TL;DR

This work analyzes the long-time behavior of small stochastic perturbations of integrable Hamiltonian systems in two complementary ways: (i) a fast-slow (action-angle) approach yielding an averaged equation for the actions, and (ii) an effective equation obtained by Krylov--Bogolyubov-type averaging in the original variables. The actions converge in distribution to the solution of the averaged dynamics under broad conditions, while an auxiliary effective equation provides a unique, globally well-behaved limit that captures the leading behavior of the actions for all times; local versions and explicit applications (chains of oscillators and damped/driven systems) illustrate the framework. The results highlight the stabilizing role of noise and friction, show when averaging yields unique limits, and contrast stochastic with deterministic perturbations, with extensions to local normal forms and potential ties to SPDEs. Overall, the paper offers a rigorous synthesis of averaging techniques for stochastic perturbations of integrable systems and clarifies when the two main approaches yield robust, time-uniform descriptions of action dynamics.

Abstract

We discuss two approaches to study the long-time behaviour and infinite-time behaviour of solutions for integrable hamiltonian systems under small stochastic perturbations. Then we compare these results with those for deterministic perturbations of integrable systems.

Paper Structure

This paper contains 11 sections, 4 theorems, 30 equations.

Table of Contents

  1. The setting
  2. First approach: the fast-slow system.
  3. Second approach: the effective equation.
  4. Local versions of the results.
  5. Applications.
  6. Chains of oscillators
  7. Damped/driven hamiltonian systems
  8. Discussion
  9. Hamiltonian perturbations of integrable systems
  10. Arbitrary perturbations of integrable systems
  11. Related results

Key Result

Theorem 2.1

The collection of curves of actions $I^\varepsilon(\tau)\ (0\le\tau \le T)$ of solutions $v^\varepsilon(\tau)$ for v_eq with $0<\varepsilon\le1$ is precompact in the space of curves, in the sense of distribution. 1) Every limiting (in the sense of distribution) random process $I^0(\tau)$ is a weak

Theorems & Definitions (7)

  • Remark 1.1
  • Theorem 2.1
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Theorem 3.4