  The wave kinetic equation predicts the averaged temporal evolution of a
continuous spectral density of waves either randomly interacting or scattered
by the fine structure of a medium. In a wide range of systems, the wave kinetic
equation is derived from a fundamental equation of wave motion, which is
symmetric through time-reversal. By contrast, the corresponding wave kinetic
equation is time-irreversible. A similar paradox appears whenever one makes a
mesoscopic description of the evolution of a very large number of microscopic
degrees of freedom. Recently, it has been understood that the kinetic
description itself, at a mesoscopic level, should not break time-reversal
symmetry. The proper theoretical or mathematical tool to derive a mesoscopic
time-reversal stochastic process is large deviation theory, for which the
deterministic wave kinetic equation appears as the most probable evolution.
This paper follows Bouchet (2020) and a series of other works that derive the
large deviation Hamiltonians of the classical kinetic theories. We propose a
derivation of the large deviation principle for the linear scattering of waves
by a weak random potential in an inhomogeneous situation. This problem involves
microscopic scales corresponding to the typical wavelengths and periods of the
waves and mesoscopic ones which are the scales of spatial inhomogeneities in
the spectral density and the time needed for the random scatterers to alter the
wave spectrum. The main assumption of the kinetic regime is a large separation
of these microscopic and mesoscopic scales. We choose a generic model of wave
scattering by weak disorder: the Schr\"odinger equation with a random
potential. We derive the path large deviation principle for the local spectral
density and discuss its main properties. We show that the mesoscopic process
obeys a time-reversal symmetry at the level of large deviations. (abridged)
