On the zero-noise limit for SDE's singular at the initial time

We investigate the zero-noise limit for SDE's driven by Brownian motion with a divergence-free drift singular at the initial time and prove that a unique probability measure concentrated on the integral curves of the drift is selected. More precisely, we prove uniqueness of the zero-noise limit for divergence-free drifts in $L^1_{loc}((0,T];BV(\mathbb{T}^d;\mathbb{R}^d))\cap L^q((0,T);L^p(\mathbb{T}^d;\mathbb{R}^d))$ where $p$ and $q$ satisfy a Prodi-Serrin condition. The vector field constructed by Depauw [C. R. Acad. Sci. Paris, 2003] lies in this class and we show that for almost every intial datum, the zero-noise limit selects a probability measure concentrated on several distinct integral curves of this vector field.