We study the well-posedness and numerical approximation of multidimensional stochastic differential equations (SDEs) with distributional drift, driven by a fractional Brownian motion. First, we prove weak existence for such SDEs. This holds under a condition that relates the Hurst parameter $H$ of the noise to the Besov regularity of the drift. Then under a stronger condition, we study the error between a solution $X$ of the SDE with drift $b$ and its tamed Euler scheme with mollified drift $b^n$. We obtain a rate of convergence in $L^m(\Omega)$ for this error, which depends on the Besov regularity of the drift. This result covers the critical case of the regime of strong existence and pathwise uniqueness. When the Besov regularity increases and the drift becomes a bounded measurable function, we recover the (almost) optimal rate of convergence $1/2-\varepsilon$. As a byproduct of this convergence, we deduce that pathwise uniqueness holds in a class of H\"older continuous solutions and that any such solution is strong. The proofs rely on stochastic sewing techniques, especially to deduce new regularising properties of the discrete-time fractional Brownian motion. We also present several examples and numerical simulations that illustrate our results.

翻译：本文研究了由分数布朗运动驱动的多维随机微分方程（SDE）在分布漂移下的适定性与数值逼近。首先，我们证明了此类SDE在弱解意义下的存在性。该结论在噪声的Hurst参数$H$与漂移项的Besov正则性满足一定关系时成立。随后在更强条件下，我们分析了具有漂移项$b$的SDE解$X$与其采用光滑化漂移$b^n$的驯服欧拉格式之间的误差。我们获得了该误差在$L^m(\Omega)$意义下的收敛速率，该速率依赖于漂移项的Besov正则性。这一结果涵盖了强解存在性与轨道唯一性准则的临界情形。当Besov正则性增强且漂移项退化为有界可测函数时，我们恢复了（近乎）最优的收敛速率$1/2-\varepsilon$。作为该收敛性的推论，我们得出在Hölder连续解类中轨道唯一性成立，且该类中的任意解均为强解。证明过程依赖于随机缝合技术，尤其在推导离散时间分数布朗运动的新正则化性质方面。我们还给出了若干实例与数值模拟以验证所得结论。