# Quasi-invariant Gaussian measures for the two-dimensional defocusing cubic nonlinear wave equation

## TL;DR

This paper proves the quasi-invariance of Gaussian measures under the two-dimensional defocusing cubic nonlinear wave equation dynamics by introducing a renormalization technique and establishing energy estimates in a probabilistic framework.

## Contribution

It introduces a novel renormalization approach to prove measure quasi-invariance for the 2D defocusing cubic NLW, advancing understanding of measure transport in nonlinear wave equations.

## Key findings

- Gaussian measures are quasi-invariant under NLW dynamics
- Renormalization of energy functional is effective in probabilistic analysis
- Establishment of energy estimates supports measure invariance results

## Abstract

We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the two-dimensional defocusing cubic nonlinear wave equation (NLW). Under some regularity condition, we prove quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces for the NLW dynamics. We achieve this goal by introducing a simultaneous renormalization on the energy functional and its time derivative and establishing a renormalized energy estimate in the probabilistic setting.