Mathematics

The works of DiPerna, Lions and Ambrosio have put on mathematical foundations the theory of flows of vector fields with one distributional spatial derivative: a single deterministic flow can be identified and it is called the regular Lagrangian flow. We argue through an example that a statistical theory is necessary for flows of vector fields when no regularity is available on the vector field. We then put forward a few questions, which should be addressed in order to develop a statistical theory of flows for arbitrary vector fields.