On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion

The problem of absolute continuity for a class of SDEs driven by a real fractional Brownian motion of any Hurst index is addressed. First, we give an elementary proof of the fact that when the diffusion coefficient does not vanish, the solution to the SDE has a positive density for all t>0t>0. Second, we extend in our setting the classical entrance-time criterion of Bouleau–Hirsch [1986. Formes de Dirichlet générales et densité des variables aléatoires réelles sur l’espace de Wiener. J. Funct. Anal. 69 (2), 229–259.]