Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions

Abstract

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.

36 pages. Accepted version

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Cited by in corpus (5)

• On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations
• On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint
• Quasi-invariance of fractional Gaussian fields nonlinear wave equation with polynomial nonlinearity
• Transport of low regularity Gaussian measures for the 1d quintic nonlinear Schrödinger equation
• Sharp quasi-invariance threshold for the cubic Szegő equation