Vol. 3, No. 2, 2022

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

Trishen Gunaratnam, Tadahiro Oh, Nikolay Tzvetkov and Hendrik Weber

Vol. 3 (2022), No. 2, 343–379

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 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 the construction of the weighted Gaussian measures, based on a variational formula for the partition function inspired by Barashkov and Gubinelli (2018), and 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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nonlinear wave equation, Gaussian measure, quasi-invariance, Euclidean quantum field theory
Mathematical Subject Classification
Primary: 35L71
Secondary: 60H30
Received: 14 September 2020
Revised: 11 June 2021
Accepted: 10 July 2021
Published: 8 July 2022
Trishen Gunaratnam
Section de mathématiques
University of Geneva
Tadahiro Oh
School of Mathematics
The University of Edinburgh
United Kingdom
Nikolay Tzvetkov
Departement of Mathematics
Université de Cergy Pontoise
Institut Universitaire de France
Cergy-Pontoise Cedex
Hendrik Weber
Department of Mathematical Sciences
University of Bath
Claverton Down
United Kingdom