Averaging principle for SDEs with singular drifts driven by α-stable processes

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 $(\alpha ,\beta )\in (0,2)\times (1-\frac{\alpha }{2},1]$. 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 $(\alpha ,\beta )\in (\frac{2}{3},1]\times (2-\frac{3\alpha }{2},1]\cup (1,2)\times (\frac{\alpha }{2},1]$. Furthermore, when $(\alpha ,\beta )\in (0,1]\times (1-\alpha ,1-\frac{\alpha }{2}]\cup (1,2)\times (\frac{1-\alpha }{2},1-\frac{\alpha }{2}]$, we show the convergence of the martingale solutions of original systems to that of the averaged equation. In particular, when $\alpha \in (1,2)$, the drift can be a distribution.