Renormalisation in the presence of variance blowup
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917875172769792 |
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| author | Hairer, Martin |
| author_facet | Hairer, Martin |
| contents | We show that if one drives the KPZ equation by the derivative of a space-time white noise smoothened out at scale $\varepsilon \ll 1$ and multiplied by $\varepsilon^{3/4}$ then, as $\varepsilon \to 0$, solutions converge to the Cole-Hopf solutions to the KPZ equation driven by space-time white noise.
In the same vein, we also show that if one drives an SDE by fractional Brownian motion with Hurst parameter $H < 1/4$, smoothened out at scale $\varepsilon \ll 1$ and multiplied by $\varepsilon^{1/4-H}$ then, as $\varepsilon \to 0$, solutions converge to an SDE driven by white noise. The mechanism giving rise to both results is the same, but the proof techniques differ substantially. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10868 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Renormalisation in the presence of variance blowup Hairer, Martin Probability 60H10, 60H15, 60L20, 60L30 We show that if one drives the KPZ equation by the derivative of a space-time white noise smoothened out at scale $\varepsilon \ll 1$ and multiplied by $\varepsilon^{3/4}$ then, as $\varepsilon \to 0$, solutions converge to the Cole-Hopf solutions to the KPZ equation driven by space-time white noise. In the same vein, we also show that if one drives an SDE by fractional Brownian motion with Hurst parameter $H < 1/4$, smoothened out at scale $\varepsilon \ll 1$ and multiplied by $\varepsilon^{1/4-H}$ then, as $\varepsilon \to 0$, solutions converge to an SDE driven by white noise. The mechanism giving rise to both results is the same, but the proof techniques differ substantially. |
| title | Renormalisation in the presence of variance blowup |
| topic | Probability 60H10, 60H15, 60L20, 60L30 |
| url | https://arxiv.org/abs/2401.10868 |