Differential and Integral Equations

Quasi-invariance of fractional Gaussian fields by the nonlinear wave equation with polynomial nonlinearity

Philippe Sosoe, William J. Trenberth, and Tianhao Xian

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Abstract

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 [15, 10].

Article information

Source
Differential Integral Equations, Volume 33, Number 7/8 (2020), 393-430.

Dates
First available in Project Euclid: 14 July 2020

Permanent link to this document
https://projecteuclid.org/euclid.die/1594692055

Mathematical Reviews number (MathSciNet)
MR4122511

Subjects
Primary: 35L71: Semilinear second-order hyperbolic equations 60H15: Stochastic partial differential equations [See also 35R60]

Citation

Sosoe, Philippe; Trenberth, William J.; Xian, Tianhao. Quasi-invariance of fractional Gaussian fields by the nonlinear wave equation with polynomial nonlinearity. Differential Integral Equations 33 (2020), no. 7/8, 393--430. https://projecteuclid.org/euclid.die/1594692055


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