Title: Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation Show Chinese title

Title: 概率适定性超越方差爆破的色散PDE I：Benjamin-Bona-Mahony方程

Abstract: We investigate a possible extension of probabilistic well-posedness theory of nonlinear dispersive PDEs with random initial data beyond variance blowup. As a model equation, we study the Benjamin-Bona-Mahony equation (BBM) with Gaussian random initial data. By introducing a suitable vanishing multiplicative renormalization constant on the initial data, we show that solutions to BBM with the renormalized Gaussian random initial data beyond variance blowup converge in law to a solution to the stochastic BBM forced by the derivative of a spatial white noise. By considering alternative renormalization, we show that solutions to the renormalized BBM with the frequency-truncated Gaussian initial data converges in law to a solution to the linear stochastic BBM with the full Gaussian initial data, forced by the derivative of a spatial white noise. This latter result holds for the Gaussian random initial data of arbitrarily low regularity. We also establish analogous results for the stochastic BBM forced by a fractional derivative of a space-time white noise. Show Chinese abstract

Abstract: 我们研究了非线性色散PDE的随机初始数据下概率适定性理论向方差爆破之外的可能扩展。 作为模型方程，我们研究了具有高斯随机初始数据的Benjamin-Bona-Mahony方程（BBM）。 通过在初始数据上引入一个合适的消失乘法重整化常数，我们证明了在方差爆破之后，具有重整化高斯随机初始数据的BBM解在分布意义上收敛到由空间白噪声导数驱动的随机BBM的解。 通过考虑其他重整化方式，我们证明了具有频率截断高斯初始数据的重整化BBM解在分布意义上收敛到由完整高斯初始数据驱动的线性随机BBM的解，该解由空间白噪声的导数驱动。 后一个结果适用于任意低正则性的高斯随机初始数据。 我们还为由时空白噪声的分数阶导数驱动的随机BBM建立了类似的结果。