Weakly nonlinear Schrödinger equation with random initial data

There is wide interest in weakly nonlinear wave equations with random initial data. A common approach is the approximation through a kinetic transport equation, which clearly poses the issue of understanding its validity in the kinetic limit. While for the general case a proof of the kinetic limit remains open, we report here on first progress. As wave equation we consider the nonlinear Schrödinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to a Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution ψt(x) of the nonlinear Schrödinger equation yields then a stochastic process stationary in x ∈ Z and t ∈ R. If λ denotes the strength of the nonlinearity, we prove that the space-time covariance of ψt(x) has a limit as λ → 0 for t = λτ , with τ fixed and |τ | sufficiently small. The limit agrees with the prediction from kinetic theory.