A non-linear wave equation with fractional perturbation

Abstract

We study a $d$-dimensional wave equation model ($2\le d\le 4$) with quadratic non-linearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter $H=(H_0,\dots ,H_d)\in (0,1{)}^{d+1}$ of the noise: if ${\sum }_{i=0}^d{H}_i>d-\frac{1}{2}$, then the equation can be treated directly, while in the case $d-\frac{3}{4}<{\sum }_{i=0}^d{H}_i\le d-\frac{1}{2}$, the model must be interpreted in the Wick sense, through a renormalization procedure. Our arguments essentially rely on a fractional extension of the considerations of \cite{gubinelli-koch-oh} for the two-dimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.

References in corpus (2)

• Random data Cauchy theory for supercritical wave equations I: Local theory
• Invariant Gibbs measures for the 2-d defocusing nonlinear wave equations

Cited by in corpus (2)

• Global dynamics for the two-dimensional stochastic nonlinear wave equations
• Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise