Asymptotic Analysis - Volume 65, issue 3-4

Authors: Camilli, Fabio | Cesaroni, Annalisa

Article Type: Research Article

Abstract: In this paper we study singular perturbations of weakly coupled systems of elliptic equations. A model problem is given by small random perturbations of random evolution processes. In this setting we give a PDE proof of large deviation results, analogous to those studied in Ann. Probab. 15 (1987), 646–658, Stochastics Stochastics Rep. 33 (1990), 111–148, Probab. Theory Related Fields 94 (1993), 335–374. An essential tool in our approach is the weak KAM theory introduced in Calc. Var. Partial Differential Equations 22 (2005), 185–228.

Keywords: weakly coupled systems, Hamilton–Jacobi equations, viscosity solutions, large deviation, random evolution process

DOI: 10.3233/ASY-2009-0944

Citation: Asymptotic Analysis, vol. 65, no. 3-4, pp. 125-146, 2009

Authors: Duyckaerts, Thomas | Fermanian Kammerer, Clotilde | Jecko, Thierry

Article Type: Research Article

Abstract: In this article, we analyze the propagation of Wigner measures of a family of solutions to a system of semi-classical pseudodifferential equations presenting eigenvalues crossings on hypersurfaces. We prove the propagation along classical trajectories under a geometric condition which is satisfied, for example, as soon as the Hamiltonian vector fields are transverse or tangent at finite order to the crossing set. We derive resolvent estimates for semi-classical Schrödinger operator with matrix-valued potential under a geometric condition of the same type on the crossing set and we analyze examples of degenerate situations where one can prove transfers between the modes.

Keywords: semi-classical Schrödinger equation, matrix-valued potential, Wigner measure, eigenvalue crossing, normal form

DOI: 10.3233/ASY-2009-0949

Citation: Asymptotic Analysis, vol. 65, no. 3-4, pp. 147-174, 2009

Authors: Sidi, Avram

Article Type: Research Article

Abstract: Let Σ∞ n=0 en [f]Pn (x) be the Legendre expansion of a function f(x) on (−1, 1). In this work, we derive an asymptotic expansion as n→∞ for en [f], assuming that f∈C∞ (−1, 1), but may have arbitrary algebraic-logarithmic singularities at one or both endpoints x=±1. Specifically, we assume that f(x) has asymptotic expansions of the forms  f(x)~Σ∞ s=0 Us l(log(1−x))(1−x)αs  as x→1−,  f(x)~Σ∞ s=0 Vs (log(1+x))(1+x)βs  as x→−1+, where Us (y) and Vs (y) are some polynomials in y. Here, αs and βs are in general complex and Rαs , Rβs …>−1. An important special case is that in which Us (y) and Vs (y) are constant polynomials; for this case, the asymptotic expansion of en [f] assumes the form  en [f]~Σs=0 αs ∉Z+ ∞ Σ∞ i=0 asi hαs +i+1/2 +(−1)n Σs=0 βs ∉Z+ ∞ Σ∞ i=0 bsi hβs +i+1/2  as n→∞, where h=(n+1/2)−2 , Z+ ={0, 1 , 2, …}, and asi and bsi are constants independent of n. Show more

Keywords: Legendre expansion, endpoint singularities, asymptotic expansions

DOI: 10.3233/ASY-2009-0950

Citation: Asymptotic Analysis, vol. 65, no. 3-4, pp. 175-190, 2009

Authors: Dostanić, Milutin R.

Article Type: Research Article

Abstract: The multipliers in the space of analytic functions with exponential mean growth are described.

Keywords: multipliers, exponential mean growth, Hadamard product

DOI: 10.3233/ASY-2009-0952

Citation: Asymptotic Analysis, vol. 65, no. 3-4, pp. 191-201, 2009

Authors: Yin, G.

Article Type: Research Article

Abstract: This work aims to developing asymptotic expansions of solutions of a system of coupled differential equations with applications to option price under regime-switching diffusions. The main motivation stems from using switching diffusions to model stochastic volatility so as to obtain uniform asymptotic expansions of European-type options. By focusing on fast mean reversion, our effort is placed on finding the “effective volatility”. Under simple conditions, asymptotic expansions are developed with uniform asymptotic error bounds. The leading term in the asymptotic expansions satisfies a Black–Scholes equation in which the mean return rate and volatility are averaged out with respect to the stationary …measure of the switching process. In addition, the full asymptotic series is developed, which will help us to gain insight on the behavior of the option price when the time approaches maturity. The asymptotic expansions obtained in this paper are interesting in their own right and can be used for other problems in control optimization of systems involving fast varying switching processes. Show more

Keywords: asymptotic expansion, fast reversion, two-time scale

DOI: 10.3233/ASY-2009-0953

Citation: Asymptotic Analysis, vol. 65, no. 3-4, pp. 203-222, 2009

Authors: Khochman, Abdallah

Article Type: Research Article

Abstract: We study the Klein paradox for the semi-classical Dirac operator on R with potentials having constant limits, possibly different at infinity. Using the complex WKB method, the time-independent scattering theory in terms of incoming and outgoing plane wave solutions is established. The corresponding scattering matrix is unitary. We obtain an asymptotic expansion, with respect to the semi-classical parameter h, of the scattering matrix in the cases of the Klein paradox, the total transmission and the total reflection. Finally, we treat the scattering problem in the zero mass case.

Keywords: semi-classical Dirac operator, scattering matrix, Klein paradox, complex WKB method

DOI: 10.3233/ASY-2009-0956

Citation: Asymptotic Analysis, vol. 65, no. 3-4, pp. 223-249, 2009

Article Type: Other

Citation: Asymptotic Analysis, vol. 65, no. 3-4, pp. 251-251, 2009