Averaging principle for SDEs with singular drifts driven by $\alpha$-stable processes
Abstract
In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with $\beta$-Hölder drift driven by $\alpha$-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 $(\alpha,\beta)\in(0,2)\times(1-\tfrac{\alpha}{2},1)$. 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 $(\alpha,\beta)\in(\tfrac{2}{3},1]\times(2-\tfrac{3\alpha}{2},1)\cup(1,2)\times(\tfrac{\alpha}{2},1)$. Furthermore, when $(\alpha,\beta)\in(0,1]\times(1-\alpha,1-\tfrac{\alpha}{2}]\cup(1,2)\times(\tfrac{1-\alpha}{2},1-\tfrac{\alpha}{2}]$, we show the convergence of the martingale solutions of original systems to that of the averaged equation. When $\alpha\in(1,2)$, the drift can be a distribution.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2024
- DOI:
- 10.48550/arXiv.2409.12706
- arXiv:
- arXiv:2409.12706
- Bibcode:
- 2024arXiv240912706C
- Keywords:
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- Mathematics - Dynamical Systems;
- Mathematics - Probability;
- 60H10;
- 34C29
- E-Print:
- 30 pages