A new approach to large deviations for the Ginzburg-Landau model

Abstract

In this work we develop stochastic control methods for the study of large deviation principles (LDP) for certain interacting particle systems. Although such methods have been well studied for analyzing large deviation properties of small noise stochastic dynamical systems [7] and of weakly interacting particle systems [6], this is the first work to implement the approach for Brownian particle systems with a local interaction. As an application of these methods we give a new proof of the large deviation principle from the hydrodynamic limit for the Ginzburg-Landau model studied in [10]. Along the way, we establish regularity properties of the densities of certain controlled Markov processes and certain results relating large deviation principles and Laplace principles in non-Polish topological spaces that are of independent interest. The proof of the LDP is based on characterizing subsequential hydrodynamic limits of controlled diffusions with nearest neighbor interaction that arise from a variational representation of certain Laplace functionals. This proof also yields a new representation for the rate function which is very natural from a control theoretic point of view. Proof techniques are very similar to those used for the law of large number analysis, namely in the proof of convergence to the hydrodynamic limit (cf. [15]). Specifically, the key step in the proof is establishing suitable bounds on relative entropies and Dirichlet forms associated with certain controlled laws. This general approach has the promise to be applicable to other interacting Brownian systems as well.