Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions
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.
Keywords
Cite
@article{arxiv.1808.03158,
title = {Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions},
author = {Trishen S. Gunaratnam and Tadahiro Oh and Nikolay Tzvetkov and Hendrik Weber},
journal= {arXiv preprint arXiv:1808.03158},
year = {2022}
}
Comments
36 pages. Accepted version