A nonlinear wave equation with fractional perturbation. (English) Zbl 1427.60121

Summary: We study a \(d\)-dimensional wave equation model (\(2\leq d\leq 4\)) 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 [M. Gubinelli et al., Trans. Am. Math. Soc. 370, No. 10, 7335–7359 (2018; Zbl 1400.35240)] for the two-dimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G22 Fractional processes, including fractional Brownian motion
35L71 Second-order semilinear hyperbolic equations


Zbl 1400.35240
Full Text: DOI arXiv Euclid


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