Quasi-invariance of fractional Gaussian fields by the nonlinear wave equation with polynomial nonlinearity. (English) Zbl 1463.35360

Sosoe, Philippe; Trenberth, William J.; Xian, Tianhao

Differ. Integral Equ. 33, No. 7-8, 393-430 (2020).

Summary: We prove quasi-invariance of Gaussian measures \(\mu_s\) with Cameron-Martin space \(H^s\) under the flow of the defocusing nonlinear wave equation with polynomial nonlinearities of any order in dimension \(d=2\) and sub-quintic nonlinearities in dimension \(d=3\), for all \(s>5/2\), including fractional \(s\). This extends work of Oh-Tzvetkov and Gunaratnam-Oh-Tzvetkov-Weber who proved this result for a cubic nonlinearity and \(s\) an even integer. The main contributions are a modified construction of a weighted measure adapted to the higher order nonlinearity, and an energy estimate for the derivative of the energy replacing the integration by parts argument introduced in previous works. We also address the question of (non) quasi-invariance for the dispersionless model raised in the introductions to [T. Oh and N. Tzvetkov, J. Eur. Math. Soc. (JEMS) 22, No. 6, 1785–1826 (2020; Zbl 1441.35175); T. S. Gunaratnam, T. Oh, N. Tzvetkov, and H. Weber, “Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions”, Preprint, arXiv:1808.03158].

MSC:

35L71	Second-order semilinear hyperbolic equations
60H15	Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

absolutely continuous; distribution; Sobolev regularity; dispersion; energy estimate

Citations:

Zbl 1441.35175