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A comprehensive proof of localization for continuous Anderson models with singular random potentials

François Germinet, Abel Klein (2013)

Journal of the European Mathematical Society

We study continuous Anderson Hamiltonians with non-degenerate single site probability distribution of bounded support, without any regularity condition on the single site probability distribution. We prove the existence of a strong form of localization at the bottom of the spectrum, which includes Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) with finite multiplicity of eigenvalues, dynamical localization (no spreading of wave packets under the time evolution),...

A lower bound for the principal eigenvalue of the Stokes operator in a random domain

V. V. Yurinsky (2008)

Annales de l'I.H.P. Probabilités et statistiques

This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound...

An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations

Antoine Gloria, Stefan Neukamm, Felix Otto (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm...

Asymptotic behaviour of stochastic semigroups.

Esther Dopazo (1990)

Extracta Mathematicae

The problem to be treated in this note is concerned with the asymptotic behaviour of stochastic semigroups, as the time becomes very large. The subject is largely motived by the Theory of Markov processes. Stochastic semigroups usually arise from pure probabilistic problems such as random walks stochastic differential equations and many others.An outline of the paper is as follows. Section one deals with the basic definitions relative to K-positivity and stochastic semigroups. Asymptotic behaviour...

Asymptotic normality of eigenvalues of random ordinary differential operators

Martin Hála (1991)

Applications of Mathematics

Boundary value problems for ordinary differential equations with random coefficients are dealt with. The coefficients are assumed to be Gaussian vectorial stationary processes multiplied by intensity functions and converging to the white noise process. A theorem on the limit distribution of the random eigenvalues is presented together with applications in mechanics and dynamics.

Asymptotic stability condition for stochastic Markovian systems of differential equations

Efraim Shmerling (2010)

Mathematica Bohemica

Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by d X ( t ) = A ( ξ ( t ) ) X ( t ) d t + H ( ξ ( t ) ) X ( t ) d w ( t ) , where ξ ( t ) is a finite-valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a necessary asymptotic stability condition.

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