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May 2019 A nonlinear wave equation with fractional perturbation
Aurélien Deya
Ann. Probab. 47(3): 1775-1810 (May 2019). DOI: 10.1214/18-AOP1296


We study a $d$-dimensional wave equation model ($2\leq d\leq4$) with quadratic nonlinearity 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 $\sum_{i=0}^{d}H_{i}>d-\frac{1}{2}$, then the equation can be treated directly, while in the case $d-\frac{3}{4}<\sum_{i=0}^{d}H_{i}\leq d-\frac{1}{2}$, 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 [Trans. Amer. Math. Soc. 370 (2017) 7335–7359] for the two-dimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.


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Aurélien Deya. "A nonlinear wave equation with fractional perturbation." Ann. Probab. 47 (3) 1775 - 1810, May 2019.


Received: 1 July 2017; Revised: 1 June 2018; Published: May 2019
First available in Project Euclid: 2 May 2019

zbMATH: 07067282
MathSciNet: MR3945759
Digital Object Identifier: 10.1214/18-AOP1296

Primary: 35L71 , 60G22 , 60H15

Keywords: Fractional noise , Stochastic wave equation , Wick renormalization

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 3 • May 2019
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