Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915787148623872 |
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| author | Butkovsky, Oleg Gallay, Samuel |
| author_facet | Butkovsky, Oleg Gallay, Samuel |
| contents | We study a multidimensional stochastic differential equation with additive noise: \[ d X_t=b(t, X_t) dt +d ξ_t, \] where the drift $b$ is integrable in space and time, and $ξ$ is either a fractional Brownian motion or a Lévy process. We show weak existence of solutions to this equation under the optimal condition on integrability indices of $b$, going beyond the subcritical Krylov--Röckner (Prodi--Serrin--Ladyzhenskaya) regime. This extends the recent results of Krylov (2020) to the fractional Brownian and Lévy cases. We also construct a counterexample to demonstrate the optimality of this condition. In the one-dimensional case, we show the existence of a strong solution under the same condition. Our methods are built upon a version of the stochastic sewing lemma of Lê and the John--Nirenberg inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_12013 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime Butkovsky, Oleg Gallay, Samuel Probability We study a multidimensional stochastic differential equation with additive noise: \[ d X_t=b(t, X_t) dt +d ξ_t, \] where the drift $b$ is integrable in space and time, and $ξ$ is either a fractional Brownian motion or a Lévy process. We show weak existence of solutions to this equation under the optimal condition on integrability indices of $b$, going beyond the subcritical Krylov--Röckner (Prodi--Serrin--Ladyzhenskaya) regime. This extends the recent results of Krylov (2020) to the fractional Brownian and Lévy cases. We also construct a counterexample to demonstrate the optimality of this condition. In the one-dimensional case, we show the existence of a strong solution under the same condition. Our methods are built upon a version of the stochastic sewing lemma of Lê and the John--Nirenberg inequality. |
| title | Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime |
| topic | Probability |
| url | https://arxiv.org/abs/2311.12013 |