Probabilistic representation and uniqueness results for measure-valued solutions of transport equations

AbstractThe Cauchy problem for a multidimensional linear non-homogeneous transport equation in divergence form is investigated. An explicit and an implicit representation formulas for the unique solution of this transport equation in the case of a regular vector field v are proved. Then, together with a regularizing argument, these formulas are used to obtain a very general probabilistic representation for measure-valued solutions in the case when the initial datum is a measure and the involved vector field is no more regular, but satisfies suitable summability assumptions w.r.t. the solution. Finally, uniqueness results for solutions of the initial-value problem are derived from the uniqueness of the characteristic curves associated to v through the theory of the probabilistic representation previously developed