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We prove quasi-invariance of Gaussian measures $\mu_s$ with Cameron-Martinspace $H^s$ under the flow of the defocusing nonlinear wave equation withpolynomial 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 $s$ aneven integer. The main contributions are a modified construction of a weightedadapted to higher order nonlinearity, and an energy estimate for the derivativeof the energy replacing the integration by parts argument introduced inprevious works. We also address the question of (non) quasi-invariance for thedispersionless model raised in the introductions to [15,10].

Quasi-invariance of fractional Gaussian fields nonlinear wave equation with polynomial nonlinearity