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

2025/09/02 by Li, Guopeng, Li, Jiawei, Oh, Tadahiro +1
#35Q35 #35R60 #60H15 #60H30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2509.02344

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.

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