  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.
