Strong existence and uniqueness for stable stochastic differential equations with distributional drift

Abstract: We consider the stochastic differential equation $$ dX_t = b(X_t) dt + dL_t,$$ where the drift $b$ is a generalized function and $L$ is a symmetric one dimensional $\alpha$-stable L\'evy processes, $\alpha \in (1, 2)$. We define the notion of solution to this equation and establish strong existence and uniqueness whenever $b$ belongs to the Besov--H\"{o}lder space $\mathcal{C}^\beta$ for $\beta >1/2-\alpha/2$.