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 in dimension $d=2$ and sub-quintic nonlinearities in dimension $d=3$, for all $s>5/2$, including fractional $s$. This extends work of Oh-Tzvetkov and Gunaratnam-Oh-Tzvetkov-Weber who proved this result for a cubic nonlinearity and $s$ an even integer. The main contributions are a modified construction of a weighted measure adapted to the 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].

中文翻译：

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分数阶非线性的非线性波动方程的分数高斯场拟不变性

我们证明了在散焦非线性波动方程的流动下，具有任意维的阶次为d $ = 2 $的多项式非线性，且为的亚五次非线性，在卡梅隆-马丁空间$ H ^ s $的情况下，证明了高斯测度$ \ mu_s $的拟不变性。尺寸$ d = 3 $，对于所有$ s> 5/2 $，包括小数$ s $。这扩展了Oh-Tzvetkov和Gunaratnam-Oh-Tzvetkov-Weber的工作，他们证明了立方非线性和$ s $为偶数整数的结果。主要贡献是改进了加权度量的构造，以适应更高阶的非线性，并且对能量导数进行了能量估计，从而代替了先前工作中引入的零件自变量积分。我们还针对[15，10]引言中提出的无色散模型，解决了（非）拟不变性的问题。