Dynamical-variational transport costs: Towards a framework for “generalised” gradient flows

Evolution equations in spaces of measures describe a wide variety of natural phenomena. The theory for such evolutions has seen tremendous growth in the last decades, of which resulted in general metric space theories for analysing variational evolutions---evolutions driven by one or more energies/entropies. On the other hand, physics and large-deviation theory suggest the study of \emph{generalised} 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 (DVTs)---a large class of large-deviation inspired functionals that provide a variational generalisation of existing transport distances---to remedy this deficiency. The role in which these objects play in the theory of \emph{generalised} gradient flows will be illustrated with an example on Markov jump processes. Finally, open questions and challenges will be mentioned.