Strong existence for SDEs with singular drifts driven by fractional Brownian motion

Abstract

Let $B^H$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in (0,1/2)$. We prove the existence of a strong solution to $$d{X}_t=b(t,X_t)dt+d{B}_t^H,X_0=x_0\in R^d,$$when $b\in L^q([0,T];L^p({\mathbb{R}}^d;{\mathbb{R}}^d)),2\le p,q<\infty$ and the subcritical Ladyzhenskaya--Prodi--Serrin condition $\frac{1}{q}+\frac{Hd}{p}<1-H$ is satisfied. Thus, within the finite-exponent range $p,q\ge 2$, the strong-existence result covers the full subcritical region, including the regime $p<2Hd$, where a uniform strong Novikov condition may fail. The proof is based on a compactness criterion for Wiener functionals. Its main ingredients are an exponentially quantitative derivative estimate for the finite-dimensional densities of fractional Brownian motion, a conditional moment estimate obtained through a martingale representation of fractional Brownian motion, and a frequency-localized truncation of the Cameron--Martin shift. These estimates yield uniform Malliavin--Sobolev bounds for smooth approximations of the equation.