Stochastic equations with singular drift driven by fractional Brownian motion

Abstract

We consider stochastic differential equation $$d{X}_t=b(X_t)dt+d{W}_t^H,$$ where the drift $b$ is either a measure or an integrable function, and $W^H$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in (0,1)$, $d\in \mathbb{N}$. For the case where $b\in L_p({\mathbb{R}}^d)$, $p\in [1,\infty ]$ we show weak existence of solutions to this equation under the condition $$\frac{d}{p}<\frac{1}{H}-1,$$ which is an extension of the Krylov-Röckner condition (2005) to the fractional case. We construct a counter-example showing optimality of this condition. If $b$ is a Radon measure, particularly the delta measure, we prove weak existence of solutions to this equation under the optimal condition $H<\frac{1}{d+1}$. We also show strong well-posedness of solutions to this equation under certain conditions. To establish these results, we utilize the stochastic sewing technique and develop a new version of the stochastic sewing lemma.

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