SDEs with subcritical Lebesgue--Hölder drift and driven by $α$-stable processes
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915520415006720 |
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| author | Tian, Rongrong Wei, Jinlong |
| author_facet | Tian, Rongrong Wei, Jinlong |
| contents | We obtain the unique weak and strong solvability for time inhomogeneous stochastic differential equations with the drift in subcritical Lebesgue--Hölder spaces $L^p([0,T];{\mathcal C}_b^β({\mathbb R}^d;{\mathbb R}^d))$ and driven by $α$-stable processes for $α\in (0,2)$. The weak well-posedness is derived for $β\in (0,1)$, $α+β>1$ and $p>α/(α+β-1)$ through Prohorov's theorem, Skorohod's representation and the regularity estimates of solutions for a class of fractional parabolic partial differential equations. The pathwise uniqueness and Davie's type uniqueness are proved for $β>1-α/2$ by using Itô--Tanaka's trick. Moreover, we give a counterexample to the pathwise uniqueness for the supercritical Lebesgue--Hölder drifts to explain the present result is sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03712 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | SDEs with subcritical Lebesgue--Hölder drift and driven by $α$-stable processes Tian, Rongrong Wei, Jinlong Probability We obtain the unique weak and strong solvability for time inhomogeneous stochastic differential equations with the drift in subcritical Lebesgue--Hölder spaces $L^p([0,T];{\mathcal C}_b^β({\mathbb R}^d;{\mathbb R}^d))$ and driven by $α$-stable processes for $α\in (0,2)$. The weak well-posedness is derived for $β\in (0,1)$, $α+β>1$ and $p>α/(α+β-1)$ through Prohorov's theorem, Skorohod's representation and the regularity estimates of solutions for a class of fractional parabolic partial differential equations. The pathwise uniqueness and Davie's type uniqueness are proved for $β>1-α/2$ by using Itô--Tanaka's trick. Moreover, we give a counterexample to the pathwise uniqueness for the supercritical Lebesgue--Hölder drifts to explain the present result is sharp. |
| title | SDEs with subcritical Lebesgue--Hölder drift and driven by $α$-stable processes |
| topic | Probability |
| url | https://arxiv.org/abs/2502.03712 |