We consider the superposition of a symmetric simple exclusion dynamics, speeded-up in time, with a spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We prove the hydrostatics and the dynamical large deviation principle.Abbreviated title (running head): LDP for a reaction-diffusion model. 2010 Mathematics Subject Classification. Primary 82C22, secondary 60F10, 82C35. Key words and phrases. Reaction-diffusion equations, hydrostatics, dynamical large deviations. 1 2 C. LANDIM, K. TSUNODA Theorem 2.2 asserts that for any ǫ > 0, µ N (V c ǫ ) vanishes as N → ∞. In contrast with previous results, equation (1.2) may not have a unique solution so that equation (1.1) may not have a global attractor, what prevents the use of the techniques developed in [21,29]. This result solves partially a conjecture raised in Subsection 4.2 of [10].The main results of this article concern the large deviations of the Glauber-Kawasaki dynamics. We first prove a full large deviations principle for the empirical measure under the sole assumption that B and D are concave functions. These assumptions encompass the case in which the potential F (ρ) = B(ρ) − D(ρ) presents two or more wells, and open the way to the investigation of the metastable behavior of this dynamics. Previous results in this directions include [14,15,3].We also prove that the large deviations rate function is lower semicontinuous and has compact level sets. These properties play a fundamental role in the proof of the static large deviation principle for the empirical measure under the stationary state µ N [9, 20].The main difficulty in the proof of the lower bound of the large deviation principle comes from the presence of exponential terms in the rate function, denoted in this introduction by I. In contrast with conservative dynamics, for a trajectory u(t, x), I(u) is not expressed as a weighted H −1 norm of ∂ t u−(1/2)∆u−F (u). This forces the development of new tools to prove that smooth trajectories are I-dense.Both the large deviations of the empirical measure under the stationary state and the metastable behavior of the dynamics in the case where the potential admits more than one well are investigated in [22] based on the results presented in this article.Comments on the proof. The proof of the law of large numbers for the empirical measure under the stationary state µ N borrows ideas from [21,29]. On the one hand, by [13], the evolution of the empirical measure is described by the solutions of the reaction-diffusion equation (1.1). On the other hand, by [12], for any density profile γ, the solution ρ t of (1.1) with initial condition γ converges to some solution of the semilinear elliptic equation (1.2). Assembling these two facts, we show in the proof of Theorem 2.2 that the empirical measure eventually reaches a neighborhood of the set of all solutions of the semilinear elliptic equation (1.2).The proof that the rate function I is lower semicontinuous and has compact level set is divided in two steps. Denote by Q(π) the energy of ...

Abstract. Let Ξ be an open and bounded subset of R d , and let F : Ξ → R be a twice continuously differentiable function. Denote by Ξ N the discretization of Ξ, Ξ N = Ξ ∩ (N −1 Z d ), and denote by X N (t) the continuous-time, nearest-neighbor, random walk on Ξ N which jumps from x to y at rate e −(1/2)N [F (y)−F (x)] . We examine in this article the metastable behavior of X N (t) among the wells of the potential F .

We study the hydrodynamic scaling limit for the Glauber-Kawasaki dynamics. It is known that, if the Kawasaki part is speeded up in a diffusive space-time scaling, one can derive the Allen-Cahn equation which is a kind of the reaction-diffusion equation in the limit. This paper concerns the scaling that the Glauber part, which governs the creation and annihilation of particles, is also speeded up but slower than the Kawasaki part. Under such scaling, we derive directly from the particle system the motion by mean curvature for the interfaces separating sparse and dense regions of particles as a combination of the hydrodynamic and sharp interface limits.

We establish the strong law of large numbers for Betti numbers of random Cech complexes built on R N -valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on R N and the case where it is supported on a C 1 compact manifold of dimension strictly less than N . The strong law is proved under very mild assumption which only requires that the common probability density function belongs to L p spaces, for all 1 ≤ p < ∞.

We consider the superposition of a symmetric simple exclusion dynamics, speeded-up in time, with a spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We prove the large deviations principle for the empirical measure under the stationary state. We deduce from this result that the stationary state is concentrated on the stationary solutions of the hydrodynamic equation which are stable.where ∆ represents the Laplacian and where B and D are non-negative polynomials.We investigate the static large deviations of the empirical measure under the stationary state. In contrast with the previous dynamics [10,21], in which the hydrodynamic equation 2010 Mathematics Subject Classification. Primary 82C22, secondary 60F10, 82C35.