  We address the problem of thermalization in the presence of a time-dependent
disorder in the framework of the nonlinear Schr\"odinger (or Gross-Pitaevskii)
equation with a random potential. The thermalization to the Rayleigh-Jeans
distribution is driven by the nonlinearity. On the other hand, the structural
disorder is responsible for a relaxation toward the homogeneous equilibrium
distribution (particle equipartition), which thus inhibits thermalization
(energy equipartition). On the basis of the wave turbulence theory, we derive a
kinetic equation that accounts for the presence of strong disorder. The theory
unveils the interplay of disorder and nonlinearity. It unexpectedly reveals
that a non-equilibrium process of condensation and thermalization can take
place in the regime where disorder effects dominate over nonlinear effects. We
validate the theory by numerical simulations of the nonlinear Schr\"odinger
equation and the derived kinetic equation, which are found in quantitative
agreement without using adjustable parameters. Experiments realized in
multimode optical fibers with an applied external stress evidence the process
of thermalization in the presence of strong disorder.
