Quasi-invariant Gaussian measures for the two-dimensional defocusing cubic nonlinear wave equation
Tadahiro Oh
The University of Edinburgh, UKNikolay Tzvetkov
Université de Cergy-Pontoise, France
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Abstract
We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the two-dimensional defocusing cubic nonlinear wave equation (NLW). Under some regularity condition, we prove quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces for the NLW dynamics. We achieve this by introducing a simultaneous renormalization of the energy functional and its time derivative and establishing a renormalized energy estimate in the probabilistic setting.
Cite this article
Tadahiro Oh, Nikolay Tzvetkov, Quasi-invariant Gaussian measures for the two-dimensional defocusing cubic nonlinear wave equation. J. Eur. Math. Soc. 22 (2020), no. 6, pp. 1785–1826
DOI 10.4171/JEMS/956