Full metastable asymptotic of the Fisher information

We establish an expansion by Gamma-convergence of the Fisher information relative to the reference measure exp(-beta V), where V is a generic multiwell potential and beta goes to infinity. The expansion reveals a hierarchy of multiple scales reflecting the metastable behavior of the underlying overdamped Langevin dynamics: distinct scales emerge and become relevant depending on whether one considers probability measures concentrated on local minima of V, probability measures concentrated on critical points of V, or generic probability measures on R^d. We thus fully describe the asymptotic behavior of minima of the Fisher information over regular sets of probabilities. The analysis mostly relies on spectral properties of diffusion operators and the related semiclassical Witten Laplacian and covers also the case of a compact smooth manifold as underlying space.Comment: 24 pages. Typos correcte

Large deviations for diffusions: Donsker-Varadhan meet Freidlin-Wentzell

We consider a diffusion process on $\mathbb R^n$ and prove a large deviation principle for the empirical process in the joint limit in which the time window diverges and the noise vanishes. The corresponding rate function is given by the expectation of the Freidlin-Wentzell functional per unit of time. As an application of this result, we obtain a variational representation of the rate function for the Gallavotti-Cohen observable in the small noise and large time limits

Spectral Analysis of Discrete Metastable Diffusions

We consider a discrete Schr\"odinger operator $H_\varepsilon= -\varepsilon^2\Delta_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb Z^d)$, where $\varepsilon>0$ is a small parameter and the potential $V_\varepsilon$ is defined in terms of a multiwell energy landscape $f$ on $\mathbb R^d$. This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of $\mathbb R^d$. It is unitarily equivalent to the generator of a diffusion on $\varepsilon \mathbb Z^d$, satisfying the detailed balance condition with respect to the Boltzmann weight $\exp{(-f/\varepsilon)}$. These type of diffusions exhibit metastable behaviour and arise in the context of disordered mean field models in Statistical Mechanics. We analyze the bottom of the spectrum of $H_\varepsilon$ in the semiclassical regime $\varepsilon\ll1$ and show that there is a one-to-one correspondence between exponentially small eigenvalues and local minima of $f$. Then we analyze in more detail the bistable case and compute the precise asymptotic splitting between the two exponentially small eigenvalues. Through this purely spectral-theoretical analysis of the discrete Witten Laplacian we recover in a self-contained way the Eyring-Kramers formula for the metastable tunneling time of the underlying stochastic process

GAMMA CONVERGENCE APPROACH FOR THE LARGE DEVIATIONS OF THE DENSITY IN SYSTEMS OF INTERACTING DIFFUSION PROCESSES

We consider extended slow-fast systems of N interacting diffusions. The typical behavior of the empirical density is described by a nonlinear McKean-Vlasov equation depending on , the scaling parameter separating the time scale of the slow variable from the time scale of the fast variable. Its atypical behavior is encapsulated in a large N Large Deviation Principle (LDP) with a rate functional. We study the Γ-convergence of as → 0 and show it converges to the rate functional appearing in the Macroscopic Fluctuations Theory (MFT) for diffusive systems