The study of evolution equations in spaces of measures has seen tremendous growth in the last decades, of which resulted in general metric space theories for analyzing variational evolutions—evolutions driven by one or more energies/entropies. On the other hand, physics and large-deviation theory suggest the study of generalized gradient flows—gradient flows with non-homogeneous dissipation potentials—which are not covered in metric space theories. In this talk, we introduce dynamical-variational transport costs—a large class of large-deviation inspired functionals that provide a variational generalization of existing transport distances—to remedy this deficiency. The role in which these objects play in the theory of generalized gradient flows will be illustrated with an example on Markov jump processes. Finally, open questions and challenges will be mentioned.

(Zoom meeting: Link – pass: 1234)