## Differential and Integral Equations

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

#### Abstract

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].

#### Article information

Source
Differential Integral Equations, Volume 33, Number 7/8 (2020), 393-430.

Dates
First available in Project Euclid: 14 July 2020