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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.12013 |
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Table of 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.