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arXiv · 1808.03158

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

Trishen S. Gunaratnam·Tadahiro Oh·Nikolay Tzvetkov·Hendrik Weber
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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.

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Trishen S. Gunaratnam, Tadahiro Oh, Nikolay Tzvetkov, Hendrik Weber. 2018-08-09. Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions. https://doi.org/10.2140/pmp.2022.3.343

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