Path large deviations for interacting diffusions with local mean-field interactions in random environment

Abstract

We consider a system of N d spins in random environment with a random local mean field type interaction. Each spin has a fixed spatial position on the torus T d , an attached random environment and a spin value in R that evolves according to a space and environment dependent Langevin dynamic. The interaction between two spins depends on the spin values, on the spatial distance and the random environment of both spins. We prove the path large deviation principle from the hydrodynamic (or local mean field McKean-Vlasov) limit and derive different expressions of the rate function for the empirical process and for the empirical measure of the paths. To this end, we generalize an approach of Dawson and Gärtner. By the space and random environment dependency, this requires new ingredients compared to mean field type interactions. Moreover, we prove the large deviation principle by using a second approach. This requires a generalisation of Varadhan's lemma to nowhere continuous functions.

Gradient flow approach to local mean-field spin systems

“…The main purpose of the Chapters 3 and 4 is to see that gradient flow methods can be used to provide elegant proofs for hydrodynamic limit results and large deviation principles. However, the representation of the rate function in Chapter 3 differs from the one in [18]. We also show here that the rate function admits a unique minimum point, which is not shown in [18].…”

“…However, the representation of the rate function in Chapter 3 differs from the one in [18]. We also show here that the rate function admits a unique minimum point, which is not shown in [18]. Moreover, in Chapter 4 we also establish the convergence in the stronger topology of the Wasserstein distance.…”

“…. , N − 1 and t ∈ [0, T ], we call θ i,N t the spin value at time t of a particle, which is located at i/N ∈ T. For a detailed historical review on such models we refer to [18,Section 1.1].…”

“…The results from Chapter 2 are new, whereas some of the results from Chapter 3 and 4 have already been proven in [4] and [18] via different approaches. For instance, the large deviation principle was proven via the approach of the paper [5] and the hydrodynamic limit was proven via the relative entropy method (see [4]).…”

The Hydrodynamic Limit for Local Mean-Field Dynamics with Unbounded Spins

“…Now, if µ N converges to some measure µ, then it is reasonable to expect that in the limit N ↑ ∞, the θ i will be independent diffusions and that their empirical distribution should, by the law of large numbers, converge to a measure µ t (dx, dθ) = ρ t (x, dθ)dx, where, for any x ∈ T d , ρ t (x, dθ) is the law of the diffusion dθ(t) = −ψ ′ (θ(t)) dt + The models we consider, and in fact an even richer class of models including random interactions and potentials, was studied from the point of view of large deviations by one of us [14] where also an extensive review of the history of these models is given. The main purpose of the present paper is to give a simple and transparent proof of just the law of large numbers (or hydrodynamic limit).…”