On the stochastic selection of integral curves of a rough vector field

Abstract We prove that for bounded, divergence-free vector fields $\boldsymbol{b}$ in $L^1_{loc}((0,1];BV(\mathbb{T}^d;\mathbb{R}^d))$ , there exists a unique incompressible measure on integral curves of $\boldsymbol{b}$ . We recall the vector field constructed by Depauw in [8], which lies in the above class, and prove that for this vector field, the unique incompressible measure on integral curves exhibits stochasticity.