# Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition (complete version)

## TL;DR

This paper develops a diagram decomposition method to analyze formal series solutions of the damped cubic NLS equation with random forcing, focusing on wave turbulence limits where amplitude diminishes and domain size grows.

## Contribution

It introduces a new diagram decomposition approach for formal series in stochastic wave turbulence analysis, extending previous work on the damped cubic NLS equation.

## Key findings

- Formal series solutions are analyzed under wave turbulence limits.
- The diagram decomposition method clarifies the structure of solutions.
- Results contribute to understanding wave turbulence in stochastic PDEs.

## Abstract

In this paper we continue to study small amplitude solutions of the damped cubic NLS equation, driven by a random force (the study was initiated in our previous work [A.Dymov, S.Kuksin, Comm. Math. Phys.'2021] and continued in [A.Dymov, S.Kuksin, A.Maiocchi, S.Vladuts, arXiv:2104.11967]). We write solutions of the equation as formal series in the amplitude and discuss the behaviour of this series under the wave turbulence limit, when the amplitude goes to zero, while the space-period goes to infinity.