Mathematics

Smirnov’s superposition principle provides an effective representation of normal 1-currents in terms of suitable measures on the space of absolutely continuous curves. This result has proved to be useful in several areas, including the analysis of PDE, optimal transport, and metric geometry. A particularly relevant application of the principle arises in the study of the continuity equation. In this setting, the theorem allows to represent solutions via measures on characteristic curves associated with the underlying ODE, known as Lagrangian representations. This approach was developed by Ambrosio within the theory of Regular Lagrangian Flows. More recently, Bianchini and Bonicatto made significant contributions by encoding the well-posedness of the continuity equation in terms of the “untangledness” property of Lagrangian representations. After an introduction to Smirnov’s superposition principle and a discussion on the above-mentioned framework, we will show how the principle extends to the even more general setting of metric spaces, as proved by Paolini and Stepanov.  Here, the notion of current considered is the one introduced by Ambrosio and Kirchheim. Finally, we will present a further extension to the case of locally normal metric currents, introduced by Lang, established in a joint work with L. Ambrosio and F. Vitillaro.