  In this work, we apply the Martin-Siggia-Rose path integral formalism to the
equations of a thin elastic plate. Using a diagrammatic technique, we obtain
the direct interaction approximation (DIA) equations to describe the evolutions
of the correlation function and the response function of the fields. Consistent
with previous results, we show that DIA equations for elastic plates can be
derived from a non-markovian stochastic process and that in the weakly
nonlinear limit, the DIA equations lead to the kinetic equation of wave
turbulence theory. We expect that this approach will allow a better
understanding of the statistical properties of wave turbulence and that DIA
equations can open new avenues for understanding the breakdown of weakly
nonlinear turbulence for elastic plates.
