We consider the lattice dynamics in the half-space. The initial data are
random according to a probability measure which enforces slow spatial variation
on the linear scale ε−1. We establish two time regimes. For
times of order ε−γ, 0<γ<1, locally the measure
converges to a Gaussian measure which is time stationary with a covariance
inherited from the initial measure (non-Gaussian, in general). For times of
order ε−1, this covariance changes in time and is governed by a
semiclassical transport equation.Comment: 35 page