  We prove quasi-invariance of Gaussian measures $\mu_s$ with Cameron-Martin
space $H^s$ under the flow of the defocusing nonlinear wave equation with
polynomial nonlinearities of any order for all $s>5/2$, including fractional
$s$. This extends work of Oh-Tzvetkov [15] and Gunaratnam-Oh-Tzvetkov-Weber
[10] who proved this result this result for a cubic nonlinearity and $s$ an
even integer. The main contributions are a modified construction of a weighted
adapted to higher order nonlinearity, and an energy estimate for the derivative
of the energy replacing the integration by parts argument introduced in
previous works. We also address the question of (non) quasi-invariance for the
dispersionless model raised in the introductions to [15,10].
