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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 |
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Vol. 3 (2022), No. 2, 343–379
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Abstract |
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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. |
Keywords
nonlinear wave equation, Gaussian measure,
quasi-invariance, Euclidean quantum field theory
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Mathematical Subject Classification
Primary: 35L71
Secondary: 60H30
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Milestones
Received: 14 September 2020
Revised: 11 June 2021
Accepted: 10 July 2021
Published: 8 July 2022
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Authors |
| Trishen Gunaratnam | |
| Section de mathématiques University of Geneva Geneva Switzerland |
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| Tadahiro Oh | |
| School of Mathematics The University of Edinburgh Edinburgh United Kingdom |
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| Nikolay Tzvetkov | |
| Departement of Mathematics Université de Cergy Pontoise Institut Universitaire de France Cergy-Pontoise Cedex France |
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| Hendrik Weber | |
| Department of Mathematical
Sciences University of Bath Claverton Down Bath United Kingdom |
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