Homogenization of an advection equation with locally stationary random coefficients

Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix

This paper deals with homogenization of second order divergence form parabolic operators with locally stationary coefficients. Roughly speaking, locally stationary coefficients have two evolution scales: both an almost constant microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims to give a macroscopic approximation that takes into account the microscopic het...

Random Fractals with Stationary Scale Coefficients

On Homogenization of Elliptic Equations with Random Coefficients

In this paper, we investigate the rate of convergence of the solution uε of the random elliptic partial difference equation (∇ε∗a(x/ε, ω)∇ε + 1)uε(x, ω) = f(x) to the corresponding homogenized solution. Here x ∈ εZd, and ω ∈ Ω represents the randomness. Assuming that a(x)’s are independent and uniformly elliptic, we shall obtain an upper bound εα for the rate of convergence, where α is a consta...

Spectral decomposition of locally stationary random processes

The notion of a locally stationary process is introduced by Silverman in [ l j . This is a new kind of a random process generalizing the notion of a weakly station­ ary process. Let {x(t)}, teR1bsa random process, generally complex, with vanishing mean value and finite covariance function R(s, t) = E{x(s) x(r)} on Wj x Mu where x(t) is the complex conjugate to x(r). The author of [ l j says tha...

Homogenization of the Schrödinger equation with large, random potential

We study the behavior of solutions to a Schrödinger equation with large, rapidly oscillating, mean zero, random potential with Gaussian distribution. We show that in high dimension d > m, where m is the order of the spatial pseudo-differential operator in the Schrödinger equation (with m = 2 for the standard Laplace operator), the solution converges in the L sense uniformly in time over finite ...