[Paper Review] A large deviation principle for Markovian slow-fast systems

Richard C. Kraaij, Mikola C. Schlottke|arXiv (Cornell University)|Nov 11, 2020

Advanced Thermodynamics and Statistical Mechanics52 references4 citations

TL;DR

This paper establishes a pathwise large deviation principle (LDP) for slow variables in weakly coupled Markovian slow-fast systems, where the time-scale separation tends to infinity. By linking the LDP to viscosity solutions of Hamilton-Jacobi-Bellman equations—whose well-posedness is established in a companion paper—it derives the rate function in two forms: a double-optimization over slow velocity and fast measure, and a principal eigenvalue problem. The key contribution is a rigorous derivation of averaging principles from large deviations, validated in a system of empirical measure-flux pairs coupled to fast diffusion on a compact manifold.

ABSTRACT

We prove pathwise large deviation principles of slow variables in slow-fast systems in the limit of time-scale separation tending to infinity. In the limit regime we consider, the convergence of the slow variable to its deterministic limit and the convergence of the fast variable to equilibrium are competing at the same scale. The large deviation principle is proven by relating the large deviation problem to solutions of Hamilton-Jacobi-Bellman equations, for which well-posedness was established in the companion paper [arXiv:1912.06579]. We cast the rate functions in action-integral form and interpret the Lagrangians in two ways. First, in terms of a double-optimization problem of the slow variable's velocity and the fast variable's distribution, similar in spirit to what one obtains from the contraction principle. Second, in terms of a principal-eigenvalue problem associated to the slow-fast system. The first representation proves in particular useful in the derivation of averaging principles from the large deviations principles. As main example of our general results, we consider empirical measure-flux pairs coupled to a fast diffusion on a compact manifold. We prove large deviations and use the Lagrangian in double-optimization form to demonstrate the validity of the averaging principle in this system.

Motivation & Objective

• To establish a pathwise large deviation principle for slow variables in weakly coupled Markovian slow-fast systems under time-scale separation.
• To relate the large deviation problem to viscosity solutions of Hamilton-Jecobi-Bellman (HJB) equations, leveraging well-posedness results from a companion paper.
• To express the rate function in two distinct forms: a double-optimization over slow velocity and fast measure distribution, and a principal eigenvalue problem.
• To demonstrate the validity of the averaging principle using the double-optimization form of the Lagrangian in a concrete example involving empirical measure-flux pairs and fast diffusion on a compact manifold.

Proposed method

• Formulate the slow-fast system as a weakly coupled Markov process with distinct time-scales, where the fast component equilibrates on the same scale as the slow component evolves.
• Apply the theory of viscosity solutions to Hamilton-Jacobi-Bellman equations to characterize the rate function of the large deviation principle.
• Derive the rate function in action-integral form, with the Lagrangian expressed as a double-optimization: over the slow variable’s velocity and the fast variable’s empirical measure distribution.
• Re-express the Lagrangian via a principal eigenvalue problem associated with the generator of the slow-fast system, providing a spectral interpretation.
• Use the double-optimization form to rigorously derive the averaging principle as a consequence of the large deviation principle.
• Validate the framework on a concrete example: empirical measure-flux pairs coupled to a fast diffusion on a compact Riemannian manifold, proving large deviations and verifying the averaging principle.

Experimental results

Research questions

• RQ1 How can a large deviation principle be rigorously established for the slow component in a weakly coupled Markovian slow-fast system when time-scale separation and equilibration occur at the same order?
• RQ2 In what ways can the rate function of the large deviation principle be represented beyond the standard action-integral form?
• RQ3 Can the double-optimization structure of the Lagrangian be used to derive the averaging principle from the large deviation framework?
• RQ4 How does the principal eigenvalue formulation of the Lagrangian relate to the dynamics of the slow-fast system?
• RQ5 To what extent does the theoretical framework apply to systems with empirical measure-flux pairs evolving under fast diffusion on a compact manifold?

Key findings

• A large deviation principle is rigorously established for the slow variable in weakly coupled Markovian slow-fast systems in the limit of infinite time-scale separation.
• The rate function is expressed in action-integral form, with the Lagrangian derived from a double-optimization over the slow velocity and the fast measure distribution, analogous to the contraction principle.
• An alternative representation of the Lagrangian is given via a principal eigenvalue problem associated with the generator of the slow-fast system, offering a spectral interpretation.
• The double-optimization form of the Lagrangian enables a direct derivation of the averaging principle from the large deviation principle, providing a rigorous justification of the averaging limit.
• In the main example—empirical measure-flux pairs coupled to fast diffusion on a compact manifold—the large deviation principle holds, and the averaging principle is validated using the double-optimization structure.
• The results are based on the well-posedness of the associated Hamilton-Jacobi-Bellman equations, which is established in the companion paper [36], ensuring the mathematical consistency of the framework.