[Paper Review] Large deviations from a stationary measure for a class of dissipative PDE's with random kicks

Vojkan Jakšić, Vahagn Nersesyan+2 more|arXiv (Cornell University)|Dec 3, 2012

Mathematical Dynamics and Fractals15 references4 citations

TL;DR

This paper establishes a large deviation principle (LDP) for occupation measures of a Markov process generated by dissipative PDEs—such as the 2D Navier-Stokes equation—perturbed by bounded, non-degenerate random kicks. Using Kifer's criterion, Lyapunov-Schmidt reduction, and asymptotic analysis of generalized Markov semigroups, the authors prove that the time-averaged empirical measures satisfy an LDP with a good rate function that is finite only on a compact set, characterizing rare events of order-one deviations from the stationary measure.

ABSTRACT

We study a class of dissipative PDE's perturbed by a bounded random kick force. It is assumed that the random force is non-degenerate, so that the Markov process obtained by the restriction of solutions to integer times has a unique stationary measure. The main result of the paper is a large deviation principle for occupation measures of the Markov process in question. The proof is based on Kifer's large deviation criterion, a Lyapunov-Schmidt type reduction, and an abstract result on large-time asymptotic for generalised Markov semigroups.

Motivation & Objective

• To establish a large deviation principle (LDP) for occupation measures of a Markov process derived from dissipative PDEs with random kick forces.
• To characterize the exponential decay rates of probabilities of large deviations of time-averaged empirical measures from the unique stationary measure.
• To extend the understanding of rare events in stochastic PDEs beyond central limit theorems and law of iterated logarithm, focusing on order-one deviations.
• To develop a general framework applicable to systems like the 2D Navier-Stokes equations and Ginzburg-Landau equations under non-degenerate random forcing.

Proposed method

• Application of Kifer's large deviation criterion to reduce the LDP problem to the analysis of long-time asymptotics of generalized Markov semigroups.
• Use of Lyapunov-Schmidt type reduction to simplify the dynamics in finite-dimensional projections, enabling control over the semigroup behavior.
• Introduction of a uniform Feller property for the Markov process to ensure regularity and convergence of transition probabilities.
• Construction of a projective limit of finite-dimensional LDPs via the inverse limit topology on the space of probability measures.
• Proof of the LDP for the full infinite-dimensional system by taking the limit of finite-dimensional approximations and verifying exponential equivalence.
• Use of the Dawson-Gärtner theorem to lift LDPs from finite-dimensional projections to the full path space of probability measures.

Experimental results

Research questions

• RQ1 What is the rate of decay for the probability that the time-averaged empirical measure of a solution to a randomly forced dissipative PDE deviates by order one from the stationary measure?
• RQ2 Can a large deviation principle be established for occupation measures of Markov processes generated by dissipative PDEs with bounded random kick forces?
• RQ3 How does the structure of the random forcing (non-degenerate, bounded, i.i.d. kicks) affect the existence and uniqueness of the stationary measure and the LDP rate function?
• RQ4 To what extent can the LDP for the full infinite-dimensional system be derived from LDPs in finite-dimensional projections?

Key findings

• The occupation measures of the Markov process satisfy a large deviation principle with a good rate function that is finite only on a compact subset of the space of probability measures.
• The rate function is characterized as the supremum over finite-dimensional projections, given by $ I(\boldsymbol{\sigma}) = \sup_{m \geq 1} I_m(\boldsymbol{\sigma} \circ p_m^{-1}) $, ensuring lower semicontinuity and compact level sets.
• The LDP holds uniformly over initial conditions, and the convergence to the stationary measure is exponentially fast in the Kantorovich-Wasserstein metric.
• The result applies to the 2D Navier-Stokes system and the Ginzburg-Landau equation under non-degenerate random kick forcing with bounded $ L^2 $-norm.
• The proof establishes the LDP via a projective limit argument, leveraging the Dawson-Gärtner theorem and uniform exponential equivalence of measures across projections.