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

We prove that for bounded, divergence-free vector fields b in L¹₋₎₂ ( (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.