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

We prove that for bounded, divergence-free vector fields b in L^1loc((0,1];BV(^d;^d)), there exists a unique incompressible measure on integral curves of b. We recall the vector field constructed by Depauw in [Depauw, C. R. Math. Acad. Sci. Paris, 2003], which lies in the above class, and prove that for this vector field, the unique incompressible measure on integral curves exhibits stochasticity.