Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911134377836544 |
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| author | Li, Guopeng Li, Jiawei Oh, Tadahiro Tzvetkov, Nikolay |
| author_facet | Li, Guopeng Li, Jiawei Oh, Tadahiro Tzvetkov, Nikolay |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation Li, Guopeng Li, Jiawei Oh, Tadahiro Tzvetkov, Nikolay Analysis of PDEs Probability 35Q35, 35R60, 60H15, 60H30 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. |
| title | Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation |
| topic | Analysis of PDEs Probability 35Q35, 35R60, 60H15, 60H30 |
| url | https://arxiv.org/abs/2509.02344 |