On the zero-noise limit for SDE's singular at the initial time
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915640878563328 |
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| author | Pitcho, Jules |
| author_facet | Pitcho, Jules |
| contents | We investigate the zero-noise limit for SDE's driven by Brownian motion with a divergence-free drift singular at the initial time and prove that a unique probability measure concentrated on the integral curves of the drift is selected. More precisely, we prove uniqueness of the zero-noise limit for divergence-free drifts in $L^1_{loc}((0,T];BV(\mathbb{T}^d;\mathbb{R}^d))\cap L^q((0,T);L^p(\mathbb{T}^d;\mathbb{R}^d))$ where $p$ and $q$ satisfy a Prodi-Serrin condition. The vector field constructed by Depauw [C. R. Acad. Sci. Paris, 2003] lies in this class and we show that for almost every intial datum, the zero-noise limit selects a probability measure concentrated on several distinct integral curves of this vector field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22905 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the zero-noise limit for SDE's singular at the initial time Pitcho, Jules Probability Analysis of PDEs Classical Analysis and ODEs 34A12, 60G99, 60H50, 76F25 We investigate the zero-noise limit for SDE's driven by Brownian motion with a divergence-free drift singular at the initial time and prove that a unique probability measure concentrated on the integral curves of the drift is selected. More precisely, we prove uniqueness of the zero-noise limit for divergence-free drifts in $L^1_{loc}((0,T];BV(\mathbb{T}^d;\mathbb{R}^d))\cap L^q((0,T);L^p(\mathbb{T}^d;\mathbb{R}^d))$ where $p$ and $q$ satisfy a Prodi-Serrin condition. The vector field constructed by Depauw [C. R. Acad. Sci. Paris, 2003] lies in this class and we show that for almost every intial datum, the zero-noise limit selects a probability measure concentrated on several distinct integral curves of this vector field. |
| title | On the zero-noise limit for SDE's singular at the initial time |
| topic | Probability Analysis of PDEs Classical Analysis and ODEs 34A12, 60G99, 60H50, 76F25 |
| url | https://arxiv.org/abs/2503.22905 |