A note on weak existence for SDEs driven by fractional Brownian motion

Abstract: We are interested in existence of solutions to the $d$-dimensional equation \begin{equation*} X_t=x_0+\int_0^t b(X_s)ds + B_t, \end{equation*} where $B$ is a (fractional) Brownian motion with Hurst parameter $H\leqslant 1/2$ and $b$ is an $\mathbb{R}^d$-valued measure in some Besov space. We exhibit a class of drifts $b$ such that weak existence holds. In particular existence of a weak solution is shown for $b$ being a finite $\mathbb{R}^d$-valued measure for any $H<1/(2d)$.