Averaging principle for SDEs with singular drifts driven by $α$-stable processes
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| Format: | Preprint |
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2024
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| _version_ | 1866917779882377216 |
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| author | Cheng, Mengyu Hao, Zimo Zhang, Xicheng |
| author_facet | Cheng, Mengyu Hao, Zimo Zhang, Xicheng |
| contents | In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with $β$-Hölder drift driven by $α$-stable processes. More specifically, we first derive the Schauder estimate for nonlocal partial differential equations (PDEs) associated with the aforementioned SDEs, within the framework of Besov-Hölder spaces. Then we consider the case where $(α,β)\in(0,2)\times(1-\tfracα{2},1)$. Using the Schauder estimate, we establish the strong convergence rate for the averaging principle. In particular, under suitable conditions we obtain the optimal rate of strong convergence when $(α,β)\in(\tfrac{2}{3},1]\times(2-\tfrac{3α}{2},1)\cup(1,2)\times(\tfracα{2},1)$. Furthermore, when $(α,β)\in(0,1]\times(1-α,1-\tfracα{2}]\cup(1,2)\times(\tfrac{1-α}{2},1-\tfracα{2}]$, we show the convergence of the martingale solutions of original systems to that of the averaged equation. When $α\in(1,2)$, the drift can be a distribution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_12706 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Averaging principle for SDEs with singular drifts driven by $α$-stable processes Cheng, Mengyu Hao, Zimo Zhang, Xicheng Dynamical Systems Probability 60H10, 34C29 In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with $β$-Hölder drift driven by $α$-stable processes. More specifically, we first derive the Schauder estimate for nonlocal partial differential equations (PDEs) associated with the aforementioned SDEs, within the framework of Besov-Hölder spaces. Then we consider the case where $(α,β)\in(0,2)\times(1-\tfracα{2},1)$. Using the Schauder estimate, we establish the strong convergence rate for the averaging principle. In particular, under suitable conditions we obtain the optimal rate of strong convergence when $(α,β)\in(\tfrac{2}{3},1]\times(2-\tfrac{3α}{2},1)\cup(1,2)\times(\tfracα{2},1)$. Furthermore, when $(α,β)\in(0,1]\times(1-α,1-\tfracα{2}]\cup(1,2)\times(\tfrac{1-α}{2},1-\tfracα{2}]$, we show the convergence of the martingale solutions of original systems to that of the averaged equation. When $α\in(1,2)$, the drift can be a distribution. |
| title | Averaging principle for SDEs with singular drifts driven by $α$-stable processes |
| topic | Dynamical Systems Probability 60H10, 34C29 |
| url | https://arxiv.org/abs/2409.12706 |