  We prove quasi-invariance of Gaussian measures supported on Sobolev spaces
under the dynamics of the three-dimensional defocusing cubic nonlinear wave
equation. As in the previous work on the two-dimensional case, we employ a
simultaneous renormalization on the energy functional and its time derivative.
Two new ingredients in the three-dimensional case are (i) the construction of
the weighted Gaussian measures, based on a variational formula for the
partition function inspired by Barashkov and Gubinelli (2018), and (ii) an
improved argument in controlling the growth of the truncated weighted Gaussian
measures, where we combine a deterministic growth bound of solutions with
stochastic estimates on random distributions.
