Lectures on random media. (English) Zbl 0814.60093

Bakry, Dominique (ed.) et al., Lectures on probability theory. Ecole d’Eté de Probabilités de Saint-Flour XXII-1992. Summer School, 9th- 25th July, 1992, Saint-Flour, France. Berlin: Springer-Verlag. Lect. Notes Math. 1581, 242-411 (1994).
This paper is a short course of lectures devoted to three central problems of the random media theory: homogenization, localization and intermittency. It is divided into three chapters each of which is concerned to one of these themes.
The first chapter is devoted to the homogenization theory, in which the initial boundary problems for stochastic partial differential equations containing ergodic random fields as coefficients are considered. The central question of the theory is the exactness of “the mean field approximation” for the expectations of random solutions of the initial boundary problems. In the lectures, the homogenization is considered for the stochastic heat propagation equation, i.e. for the disordered solid state model, for the plasma turbulent transport model and for the random walk model in random media.
The second chapter is devoted to the localization problem which consists of the identification of the pure point component in the spectrum of random one-particle quantum Hamiltonians \(H (\omega)\), \(\omega \in \Omega\), with random potential. This problem is considered for the one- dimensional model with Hamiltonians \[ H (\omega) = - d^ 2/dx^ 2 + V(x, \omega) \] in \(L_ 2 (\mathbb{R})\) and for many-dimensional lattice Hamiltonians \[ H(\omega) \psi = \sum_{x' : | x' - x | = 1} \bigl( \psi (x') - \psi (x) \bigr) + \sigma V(x, \omega), \quad x \in \mathbb{Z}^ d, \;\omega \in \Omega, \;d \geq 1. \] The same localization problem is considered for the stochastic wave equation of random strings.
The third chapter is concerned to the intermittency property of the evoluting or stationary random fields. From the qualitative viewpoint, the intermittent random fields form the strong rare random spatial fluctuations giving the main contribution to some physical processes in random media. The simple formulation of the intermittency property of the time random field evolution is the following one. Let \(u(t,x)\), \(t \in [0, \infty)\), \(x \in \mathbb{Z}^ d\), be a family of nonnegative, homogeneous random fields which are ergodic in space \(\mathbb{Z}^ d\) and let \[ 0 < \sup \lim \ln \mathbb{E} u^ p (t,x) = \gamma_ p < \infty, \quad p \in \mathbb{N} \] (the left inequality takes place for sufficiently large \(p\)). The family \(u(t,x)\) is asymptotically intermittent if the strict inequalities \(\gamma_ 2/2<\gamma_ 3/3<\gamma_ 4/4< \dots\) are fulfilled. The intermittency problem is considered for the parabolic Anderson model with random potential and for the cell-dynamo model in the magnetohydrodynamics.
For the entire collection see [Zbl 0797.00021].


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60-02 Research exposition (monographs, survey articles) pertaining to probability theory