Asymptotic Analysis

For sequences of non-lattice weakly dependent random variables, we obtain asymptotic expansions for Large Deviation Principles. These expansions, commonly referred to as strong large deviation results, are in the spirit of Edgeworth Expansions for the Central Limit Theorem. We show that the results are applicable to Diophantine iid sequences, finite state Markov chains, strongly ergodic Markov chains and Birkhoff sums of smooth expanding maps & subshifts of finite type.

We establish a convergence theorem for a class of two components nonlinear reaction–diffusion systems. Each diffusion term is the subdifferential of a convex functional of the calculus of variations whose class is equipped with the Mosco-convergence. The reaction terms are structured in such a way that the systems admit bounded solutions, which are positive in the modeling of ecosystems. As a consequence, under a stochastic homogenization framework, we prove two homogenization theorems for this class. We illustrate the results with the stochastic homogenization of a prey–predator model with saturation effect.

Starting from a complete family for the unit sphere S n in the complex n-space C n (whose elements are coherent states attached to the Barut–Girardello space), we obtain an asymptotic expansion for the associated Berezin transform. The proof involves the computation of the asymptotic behaviour of functions in the complete family. Furthermore, in an analogous manner (slightly weaker) we obtain the asymptotic expansion of the covariant symbol of a pseudo-differential operator on L 2 ( S n ) .

Let α ∈ ] 0 , 1 [ . Let Ω o be a bounded open domain of R n of class C 1 , α . Let ν Ω o denote the outward unit normal to ∂ Ω o . We assume that the Steklov problem Δ u = 0 in Ω o , ∂ u ∂ ν Ω o = λ u on ∂ Ω o has a multiple eigenvalue λ ˜ of multiplicity r. Then we consider an annular domain Ω ( ϵ ) obtained by removing from Ω o a small cavity of class C 1 , α and size ϵ > 0 , and we show that under appropriate assumptions each elementary symmetric function of r eigenvalues of the Steklov problem Δ u = 0 in Ω ( ϵ ) , ∂ u ∂ ν Ω ( ϵ ) = λ u on ∂ Ω ( ϵ ) which converge to λ ˜ as ϵ tend to zero, equals real a analytic function defined in an open neighborhood of ( 0 , 0 ) in R 2 and computed at the point ( ϵ , δ 2 , n ϵ log ϵ ) for ϵ > 0 small enough. Here ν Ω ( ϵ ) denotes the outward unit normal to ∂ Ω ( ϵ ) , and δ 2 , 2 ≡ 1 and δ 2 , n ≡ 0 if n ⩾ 3 . Such a result is an extension to multiple eigenvalues of a previous result obtained for simple eigenvalues in collaboration with S. Gryshchuk.

In previous work the authors found the asymptotic expansion of the L 2 -norm of the solution u ( t , x ) of the strongly damped wave equation u t t − Δ u t − Δ u = 0 and also of the L 2 -norm of the difference between u ( t , x ) and its asymptotic approximation ν ( t , x ) . This was done in space dimension N ⩾ 3 . In the present work results are extended to the exceptional cases N = 1 and N = 2 . This extension is achieved by deriving new lemmas on the asymptotic expansion of some parameter dependent integrals.

We are concerned with the analysis of the approximation by diffusion and homogenization of a Vlasov–Poisson–Fokker–Planck system. Here we generalize the convergence result of (Comm. Math. Sci. 8 (2010), 463–479) where the same problem is treated without the oscillating electrostatic potential and we extend the one dimensional result of (Ann. Henri Poincaré 17 (2016), 2529–2553) to the case of several space dimensions. An averaging lemma and two scale convergence techniques are used to prove rigorously the convergence of the scaled Vlasov–Poisson–Fokker–Planck system to a homogenized Drift-Diffusion-Poisson system.