Sobolev functions on spaces of measures and applications

After a brief introduction to the Wasserstein and Hellinger-Kantorovich distances on the space of (probability) measures, we discuss two approaches to the definition of differentiable functions on spaces of measures. Provided a suitable (and in a way canonical) reference measure is chosen, we show that the two approaches coincide. Finally, we discuss some possible applications such as the study of HJB equations and stochastic processes on spaces of measures.