Title: Wick polynomials and time-evolution of cumulants

Abstract:

We show how Wick polynomials of random variables can be defined 
combinatorially as the unique choice which removes all ``internal
contractions'' from the related cumulant expansions, also in a 
non-Gaussian case.
We discuss how an expansion in terms of the Wick polynomials can be used for
derivation of a hierarchy of equations for the time-evolution of cumulants.
These methods are then applied to simplify the formal derivation of
the Boltzmann-Peierls equation in the kinetic scaling limit of the discrete 
nonlinear Sch\"{o}dinger equation (DNLS) with suitable random
initial data.  We also present a reformulation of the standard perturbation
expansion using cumulants which could simplify the problem of a rigorous 
derivation of the Boltzmann-Peierls equation by separating the analysis of 
the solutions to the Boltzmann-Peierls equation from the analysis of the 
corrections.  This latter scheme is general and not tied to the DNLS 
evolution equations.