We prove the hydrodynamical limit for weakly asymmetric simple exclusion processes. A large deviation property with respect to this limit is established for the symmetric case. We treat also the situation where a slow reaction (creation and annihilation of particles) is present. IntroductionWhen we study the hydrodynamical limit (reduced description) of an infinite particle system we can obtain the same limiting equation for different systems. For instance, both symmetric simple exclusion and a system of independent symmetric random walks give rise to the heat equation. However, in the study of the large deviations, the rate functions are different for these two systems.In this spirit the large deviations from the hydrodynamical limit for independent Brownian motions were studied in [7], using an approach that can be also used for strongly interacting systems, like simple exclusion, as we show in this paper.The case of independent particles is simple because only the empirical density field appears in the martingales considered. In the case of strongly interacting systems the density field is no longer autonomous and we need to control the deviations of the empirical correlation fields. In Theorem 2.1 we prove that the probability that any correlation field deviates from an appropriate function of the density field is superexponentially small, whereas the large deviations of the density fields are only exponentially small.The other ingredient needed for the lower bound is a large number of hydrodynamical limit theorems for several small perturbations of the original

Anomalous large thermal conductivity has been observed numerically and experimentally in one- and two-dimensional systems. There is an open debate about the role of conservation of momentum. We introduce a model whose thermal conductivity diverges in dimensions 1 and 2 if momentum is conserved, while it remains finite in dimension d > or = 3. We consider a system of harmonic oscillators perturbed by a nonlinear stochastic dynamics conserving momentum and energy. We compute explicitly the time correlation function of the energy current C(J)(t), and we find that it behaves, for large time, like t(-d/2) in the unpinned cases, and like t(-d/2-1) when an on-site harmonic potential is present. This result clarifies the role of conservation of momentum in the anomalous thermal conductivity in low dimensions.

We study the large scale space-time fluctuations of an interface which is modeled by a massless scalar field with reversible Langevin dynamics. For a strictly convex interaction potential we prove that on a large space-time scale these fluctuations are governed by an infinite-dimensional Ornstein-Uhlenbeck process. Its effective diffusion type covariance matrix is characterized through a variational formula.

We study the hyperbolic scaling limit for a chain of N coupled anharmonic oscillators. The chain is open and with the following adiabatic boundary conditions: it is attached to a wall on the left and there is a force (tension) τ acting on the right. In order to provide the system of the good ergodic properties, we perturb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that momentum and energy are locally conserved. We prove that in the macroscopic limit the distribution of the density of particles, momentum and energy converge to the solution of the Euler equations, in the smooth regime of them.

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We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute thermal conductivity via Green-Kubo formula. In the harmonic case we compute the current-current time correlation function, that decay like t −d/2 in the unpinned case and like t −d/2−1 if a on-site harmonic potential is present. This implies a finite conductivity in d ≥ 3 or in pinned cases, and we compute it explicitely. For general anharmonic strictly convex interactions we prove some upper bounds for the conductivity that behave qualitatively as in the harmonic cases.Date: October 12, 2018.

Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability measure $\pi$. Let $\Psi$ a function on the state space of the chain, with $\alpha$-tails with respect to $\pi$, $\alpha\in (0,2)$. We find sufficient conditions on the probability transition to prove convergence in law of $N^{1/\alpha}\sum_n^N \Psi(X_n)$ to a $\alpha$-stable law. ``Martingale approximation'' approach and ``coupling'' approach give two different sets of conditions. We extend these results to continuous time Markov jump processes $X_t$, whose skeleton chain satisfies our assumptions. If waiting time between jumps has finite expectation, we prove convergence of $N^{-1/\alpha}\int_0^{Nt} V(X_s) ds$ to a stable process. In the case of waiting times with infinite average, we prove convergence to a Mittag-Leffler process.Comment: Accepted for the publication in Annals of Applied Probabilit

We consider lattice dynamics with a small stochastic perturbation of order ε and prove that for a space-time scale of order ε −1 the local spectral density (Wigner function) evolves according to a linear transport equation describing inelastic collisions. For an energy and momentum conserving chain the transport equation predicts a slow decay, as 1/ √ t, for the energy current correlation in equilibrium. This is in agreement with previous studies using a different method.Proof. We use the evolution equation (33) where K is the bound on the total energy from condition (b3). Then by Gronwall's inequality J ρ , E ε (t) ≤ 2ρK + J ρ , E ε (0) e c 3 γt ,