[Paper Review] Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging

Yuri Kifer|arXiv (Cornell University)|Oct 12, 2007

Mathematical Dynamics and Fractals56 references5 citations

TL;DR

This paper establishes large deviations principles for slow-fast dynamical systems and Markov processes in the fully coupled averaging regime, where fast motions depend on slow variables. It derives both upper and lower large deviation bounds, enabling analysis of rare adiabatic transitions between attractors of the averaged system on exponentially long time scales, extending classical averaging beyond the uncoupled case.

ABSTRACT

The work treats systems combining slow and fast motions depending on each other where fast motions are perturbations of families of either dynamical systems or Markov processes with freezed slow variable. In the first case we consider hyperbolic dynamical systems and in the second case we deal with random evolutions which are combinations of diffusions and continuous time Markov chains. We study first large deviations of the slow motion from the averaged one and then use these results together with some Markov property type arguments in order to describe very long time behavior of the slow motion such as its transitions between attractors of the averaged system.

Motivation & Objective

• To extend the classical averaging principle to the fully coupled case where fast motions depend on slow variables, which is more realistic but mathematically complex.
• To establish both upper and lower large deviation bounds for the slow motion $X^\varepsilon$ from its averaged limit $\bar{X}^\varepsilon$, going beyond convergence in measure or probability.
• To analyze the long-time behavior of the slow motion on exponentially long time scales, particularly rare transitions between basins of attraction of the averaged system.
• To develop a framework for studying fluctuations and adiabatic transitions in systems with hyperbolic fast dynamics and Markov processes such as diffusions and continuous-time Markov chains.
• To provide a rigorous foundation for modeling phenomena like climate–weather interactions and perturbed Hamiltonian systems where coupling between time scales is intrinsic.

Proposed method

• Use of large deviation theory to analyze fluctuations of the slow motion $X^\varepsilon$ around its averaged limit $\bar{X}^\varepsilon$, deriving both upper and lower bounds via rate functions.
• Application of the $S$-functional formalism to characterize the action functional governing large deviations, leveraging Markov property arguments in the hyperbolic case and exact Markov property in the Markov process case.
• Construction of a time-scale separation via a splitting of the slow motion into intervals, using a partitioning of the time horizon and empirical measures to control deviations.
• Employment of Young measures and ergodic theory to handle weak limits and invariant measures $\mu_x$ of the fast flow $F^t_x$, ensuring the averaged vector field $\bar{B}(x)$ is well-defined.
• Use of coupling arguments and concentration inequalities to bound the difference between the actual slow motion and its averaged version, relying on Lipschitz continuity and hyperbolicity of fast dynamics.
• Extension of results to discrete-time difference equations and random evolutions, including stochastic resonance as a special case, via analogous large deviation and averaging arguments.

Experimental results

Research questions

• RQ1 How can the classical averaging principle be extended to systems where fast motions depend on slow variables (fully coupled case), rather than being independent?
• RQ2 What large deviation principles govern the fluctuations of the slow motion $X^\varepsilon$ from its averaged limit $\bar{X}^\varepsilon$ on the time scale $1/\varepsilon$?
• RQ3 What is the behavior of the slow motion on exponentially long time scales $\exp(1/\varepsilon)$, particularly regarding rare transitions between attractors of the averaged system?
• RQ4 How do the large deviation rate functions and action functionals characterize adiabatic transitions between basins of attraction in the fully coupled setting?
• RQ5 Under what conditions does the averaged system accurately describe the long-term stochastic behavior of the full system, especially when fast motions are Markov processes?

Key findings

• Both upper and lower large deviation bounds are established for the slow motion $X^\varepsilon$ in the fully coupled case, enabling a full Freidlin-Wentzell-type large deviation principle.
• The large deviation rate function is characterized via an action integral involving the averaged vector field $\bar{B}(x)$ and the invariant measure $\mu_x$ of the fast motion.
• On exponentially long time scales, the slow motion exhibits rare adiabatic transitions between neighborhoods of attractors of the averaged system, governed by the large deviation rate function.
• The averaging principle in the form $\varepsilon \int E[\sup_t |X^\varepsilon(t) - \bar{X}^\varepsilon(t)|] \to 0$ is strengthened to a uniform convergence over all bounded continuous functions $f$, under suitable ergodicity and regularity conditions.
• For Markov fast motions such as diffusions or continuous-time Markov chains, the large deviation bounds hold under conditions ensuring existence of smooth, positive invariant densities $q(x,y)$ and ergodicity for $\lambda$-a.e. $x$.
• The results extend to discrete-time difference equations and random evolutions, with analogous large deviation and averaging results derived via similar techniques.