Asymptotic Analysis - Volume 121, issue 3-4

Authors: Fernando, Kasun | Hebbar, Pratima

Article Type: Research Article

Abstract: 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.

Keywords: Large deviations, asymptotic expansions, weak dependence, Markov chains, expanding maps, subshifts of finite type

DOI: 10.3233/ASY-201602

Citation: Asymptotic Analysis, vol. 121, no. 3-4, pp. 219-257, 2021

Authors: Anza Hafsa, Omar | Mandallena, Jean Philippe | Michaille, Gérard

Article Type: Research Article

Abstract: 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.

Keywords: Convergence of two components reaction–diffusion equations, stochastic homogenization, prey–predator models

DOI: 10.3233/ASY-201603

Citation: Asymptotic Analysis, vol. 121, no. 3-4, pp. 259-305, 2021

Authors: Díaz-Ortíz, Erik I.

Article Type: Research Article

Abstract: 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 ) …. Show more

Keywords: Holomorphic space, Berezin transform, Bessel function, asymptotic approximation, semiclassical pseudo-differential operator, coherent states

DOI: 10.3233/ASY-201604

Citation: Asymptotic Analysis, vol. 121, no. 3-4, pp. 307-333, 2021

Authors: Lanza de Cristoforis, Massimo

Article Type: Research Article

Abstract: 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. Show more

Keywords: Multiple Steklov eigenvalues and eigenfunctions, singularly perturbed domain, Laplace operator, real analytic continuation

DOI: 10.3233/ASY-201605

Citation: Asymptotic Analysis, vol. 121, no. 3-4, pp. 335-365, 2021

Authors: Barrera, Joseph | Volkmer, Hans

Article Type: Research Article

Abstract: 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. Show more

Keywords: Asymptotic analysis, asymptotic expansion of L2-norm, wave equation, strong damping, Fourier analysis, weighted L1 initial data

DOI: 10.3233/ASY-201606

Citation: Asymptotic Analysis, vol. 121, no. 3-4, pp. 367-399, 2021

Authors: Addala, Lanoir | El Ghani, Najoua | Tayeb, Mohamed Lazhar

Article Type: Research Article

Abstract: 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.

Keywords: Vlasov–Poisson–Fokker–Planck system, Drift-Diffusion equation, diffusion approximation, homogenization, renormalized solution, two scale convergence, averaging lemma

DOI: 10.3233/ASY-201608

Citation: Asymptotic Analysis, vol. 121, no. 3-4, pp. 401-422, 2021