A non-linear wave equation with fractional perturbation

TL;DR

This paper investigates a stochastic wave equation with quadratic non-linearity and fractional noise, identifying regimes where the equation is well-posed directly or requires Wick renormalization, extending previous fractional noise analyses.

Contribution

It extends the analysis of stochastic wave equations with fractional noise to higher dimensions and different regimes, providing a fractional framework for renormalization and well-posedness.

Findings

Direct treatment possible when sum of Hurst parameters exceeds d-1/2

Wick renormalization needed when sum of Hurst parameters is between d-3/4 and d-1/2

Framework generalizes previous white-noise and polynomial perturbation results

Abstract

We study a $d$-dimensional wave equation model ($2\leq d\leq 4$) with quadratic non-linearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter $H=(H_0,\ldots,H_d) \in (0,1)^{d+1}$ of the noise: if $\sum_{i=0}^d H_i > d-\frac12$, then the equation can be treated directly, while in the case $d-\frac34<\sum_{i=0}^d H_i\leq d-\frac12$, 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.