Effective media for periodic anisotropic systems

One considers the wave propagation in periodic anisotropic systems. For these systems, with the aid of a matrix method, one determines the effective media.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 130–138, 1983.     

Lifshitz tails for random perturbations of periodic Schrödinger operators

The present paper is a non-exhaustive review of Lifshitz tails for random perturbations of periodic Schrödinger operators. It is not our goal to review the whole literature on Lifshitz tails; we will concentrate on a single model, the continuous Anderson model.     

Bernstein inequality under averaging of elliptic systems in periodic random media

We construct the exponential Bernstein inequality for normed fluctuations of a solution of the Dirichlet problem with rapidly oscillating periodic random coefficients with respect to a solution of the averaged Dirichlet problem.     

2-dimensional localization of acoustic waves in random perturbation of periodic media

In this work we study internal band edges localization of acoustic waves in 2-dimensional space obtained by random perturbation of some periodic media. Our results rely on the study of Lifshitz tails for the integrated density of states for random acoustic operators of the form . Localization is then deduced by the standard multiscale argument.     

Periodic orbits for anisotropic perturbations of the Kepler problem

We study the Kepler problem perturbed by an anisotropic term, that is a potential conformed by a Newtonian term, 1/r$1/r$, plus an anisotropic term, b/(r²[1+?cos²θ])^β/2$b/{\left(r^2[1+?{cos}^2\theta ]\right)}^{\beta /2}$. Because of the anisotropic term, although the system is conservative the angular momentum is not a constant of motion.     

Covolume techniques for anisotropic media

Summary The covolume method, a new approach applicable on general meshes, is extended to discretize and numerically solve the div-curl system in anisotropic media. The covolume method gives simple schemes and good approximations to the solution of the div-curl system. It works directly with the system and utilizes dual pairs of meshes that are orthogonally related. Central to the approach is the introduction of field components tangent and normal to the edges of one of the meshes, and the employment of dual discretization on the dual mesh pairs. The discretization procedures, schemes and error analysis are presented. The convergence of the method is proved.The work was partially done while this author was at Carnegie Mellon University     

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

We consider a periodic self-adjoint pseudo-differential operator H=(?Δ) ^m +B, m>0, in ? ^d which satisfies the following conditions: (i) the symbol of B is smooth in x, and (ii) the perturbation B has order less than 2m. Under these assumptions, we prove that the spectrum of H contains a half-line. This, in particular implies the Bethe-Sommerfeld conjecture for the Schrödinger operator with a periodic magnetic potential in all dimensions.     

Localization for branching random walks in random environment

We consider branching random walks in d$d$-dimensional integer lattice with time–space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If d≥3$d\ge 3$ and the environment is “not too random”, then, the total population grows as fast as its expectation with strictly positive probability. If, on the other hand, d≤2$d\le 2$, or the environment is “random enough”, then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of “replica overlap”. We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.     

Metastability for nonlinear random perturbations of dynamical systems

In this paper we describe the long time behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of the corresponding diffusion processes.     

Some singular perturbations of periodic operators

We consider two examples of singular perturbations of a periodic differential operator on the axis. The first example is a rapidly oscillating potential with compact support, and the second example is the delta potential with a small complex coupling constant. We investigate the structure and asymptotic behavior of the spectrum of the perturbed operators in detail. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 207–218, May, 2007.     

Discrete spectrum for complex perturbations of periodic Jacobi matrices

We study spectrum inclusion regions for complex Jacobi matrices, which are compact perturbations of real periodic Jacobi matrix. The condition sufficient for the lack of the discrete spectrum for such matrices is given.     

Resonances for periodic Jacobi operators with finitely supported perturbations

We describe the resonances and the eigenvalues of a periodic Jacobi operator with finitely supported perturbations. In the case of small diagonal perturbations we determine their asymptotics.     

The periodic problem for curvature-like equations with asymmetric perturbations

We discuss existence and multiplicity of solutions of the periodic problem for the curvature-like equation

     

Local Whittle estimator for anisotropic random fields

A local Whittle estimator is developed to simultaneously estimate the long memory parameters for stationary anisotropic scalar random fields. It is shown that these estimators are consistent and asymptotically normal, under some weak technical conditions. A brief simulation study illustrates a practical application of the estimator.     

Collapse of attractors for ODEs under small random perturbations

This paper concerns comparisons between attractors for random dynamical systems and their corresponding noiseless systems. It is shown that if a random dynamical system has negative time trajectories that are transient or explode with probability one, then the random attractor cannot contain any open set. The result applies to any Polish space and when applied to autonomous stochastic differential equations with additive noise requires only a mild dissipation of the drift. Additionally, following observations from numerical simulations in a previous paper, analytical results are presented proving that the random global attractors for a class of gradient-like stochastic differential equations consist of a single random point. Comparison with the noiseless system reveals that arbitrarily small non-degenerate additive white noise causes the deterministic global attractor, which may have non-zero dimension, to ‘collapse’. Unlike existing results of this type, no order preserving property is necessary.      

The central limit theorem for random perturbations of rotations

Summary.   We prove a functional central limit theorem for stationary random sequences given by the transformations

Polar sets for anisotropic Gaussian random fields

This paper studies polar sets for anisotropic Gaussian random fields, i.e. sets which a Gaussian random field does not hit almost surely. The main assumptions are that the eigenvalues of the covariance matrix are bounded from below and that the canonical metric associated with the Gaussian random field is dominated by an anisotropic metric. We deduce an upper bound for the hitting probabilities and conclude that sets with small Hausdorff dimension are polar. Moreover, the results allow for a translation of the Gaussian random field by a random field, that is independent of the Gaussian random field and whose sample functions are of bounded Hölder norm.