Weak well-posedness of energy solutions to singular SDEs with supercritical distributional drift
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| Format: | Preprint |
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2024
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| author | Gräfner, Lukas Perkowski, Nicolas |
| author_facet | Gräfner, Lukas Perkowski, Nicolas |
| contents | We study stochastic differential equations with additive noise and distributional drift on $\mathbb{T}^d$ or $\mathbb{R}^d$ and $d \geqslant 2$. We work in a scaling-supercritical regime using energy solutions and recent ideas for generators of singular stochastic partial differential equations. We mainly focus on divergence-free drift, but allow for scaling-critical non-divergence free perturbations. In the time-dependent divergence-free case we roughly speaking prove weak well-posedness of energy solutions with initial law $μ\ll \text{Leb}$ for drift $b \in L^p_T B^{-γ}_{p, 1}$ with $p \in (2, \infty]$ and $p \geqslant \frac{2}{1 -γ}$. For time-independent $b$ we show weak well-posedness of energy solutions with initial law $μ\ll \text{Leb}$ under certain structural assumptions on $b$ which allow local singularities such that $b \notin B^{-1}_{2 d/(d-2), 2}$, meaning that for any $p > 2$ in sufficiently high dimension there exists $b \notin B^{-1}_{p, 2}$ such that weak well-posedness holds for energy solutions with drift $b$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_09046 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak well-posedness of energy solutions to singular SDEs with supercritical distributional drift Gräfner, Lukas Perkowski, Nicolas Probability Analysis of PDEs We study stochastic differential equations with additive noise and distributional drift on $\mathbb{T}^d$ or $\mathbb{R}^d$ and $d \geqslant 2$. We work in a scaling-supercritical regime using energy solutions and recent ideas for generators of singular stochastic partial differential equations. We mainly focus on divergence-free drift, but allow for scaling-critical non-divergence free perturbations. In the time-dependent divergence-free case we roughly speaking prove weak well-posedness of energy solutions with initial law $μ\ll \text{Leb}$ for drift $b \in L^p_T B^{-γ}_{p, 1}$ with $p \in (2, \infty]$ and $p \geqslant \frac{2}{1 -γ}$. For time-independent $b$ we show weak well-posedness of energy solutions with initial law $μ\ll \text{Leb}$ under certain structural assumptions on $b$ which allow local singularities such that $b \notin B^{-1}_{2 d/(d-2), 2}$, meaning that for any $p > 2$ in sufficiently high dimension there exists $b \notin B^{-1}_{p, 2}$ such that weak well-posedness holds for energy solutions with drift $b$. |
| title | Weak well-posedness of energy solutions to singular SDEs with supercritical distributional drift |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2407.09046 |