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

Trishen Gunaratnam, Tadahiro Oh, Nikolay Tzvetkov, Hendrik Weber

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

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.
Original languageEnglish
Pages (from-to)343-379
Number of pages37
JournalProbability and Mathematical Physics
Issue number2
Publication statusPublished - 9 Jul 2022

Keywords / Materials (for Non-textual outputs)

  • nonlinear wave equation
  • Gaussian measure
  • quasi-invariance
  • Euclidean quantum field theory


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