A NONLINEAR WAVE EQUATION WITH FRACTIONAL PERTURBATION

• 成果类型: Article
• 署名作者: Deya, Aurelien
• 署名单位: Universite de Lorraine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1296
• 发表日期: 2019
• 页码: 1775-1810
• 关键词: feynman-kac formula; driven; noise

摘要

We study a d-dimensional wave equation model (2 <= d <= 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, ... , H-d) is an element of (0, 1)(d+1) of the noise: If Sigma(d)(i=0) H-i > d - 1/2, then the equation can be treated directly, while in the case d - 3/4 < Sigma(d)(i=0) H-i <= d - 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.