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.