Stochastic flows for Hölder drifts and transport/continuity equations with noise
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914068727595008 |
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| author | Ørke, Magnus C. |
| author_facet | Ørke, Magnus C. |
| contents | We prove existence of a stochastic flow of diffeomorphisms generated by SDEs with drift in $L^q_t C^{0, α}_x$ for any $q \in [2, \infty)$ and $α\in (0, 1)$. This result is achieved using a Zvonkin-type transformation for the SDE. As a key intermediate step, well-posedness and optimal regularity for a class of parabolic PDEs related to the transformation is established. Using the existence of a differentiable stochastic flow, we prove well-posedness of $BV_\text{loc}$-solutions of stochastic transport equations and weak solutions of stochastic continuity equations with so-called transport noise and velocity fields in $L^q_t C^{0, α}_x$. For both equations, solutions may fail to be unique in the deterministic setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01759 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic flows for Hölder drifts and transport/continuity equations with noise Ørke, Magnus C. Probability Analysis of PDEs 60H10, 60H15, 35B65 We prove existence of a stochastic flow of diffeomorphisms generated by SDEs with drift in $L^q_t C^{0, α}_x$ for any $q \in [2, \infty)$ and $α\in (0, 1)$. This result is achieved using a Zvonkin-type transformation for the SDE. As a key intermediate step, well-posedness and optimal regularity for a class of parabolic PDEs related to the transformation is established. Using the existence of a differentiable stochastic flow, we prove well-posedness of $BV_\text{loc}$-solutions of stochastic transport equations and weak solutions of stochastic continuity equations with so-called transport noise and velocity fields in $L^q_t C^{0, α}_x$. For both equations, solutions may fail to be unique in the deterministic setting. |
| title | Stochastic flows for Hölder drifts and transport/continuity equations with noise |
| topic | Probability Analysis of PDEs 60H10, 60H15, 35B65 |
| url | https://arxiv.org/abs/2501.01759 |