Evolution variational inequalities with general costs

We extend the theory of gradient flows beyond metric spaces by studying evolution variational inequalities (EVIs) driven by general cost functions c, including Bregman and entropic transport divergences. We establish several properties of the resulting flows for semiconvex functionals, including stability and energy identities. Using novel notions of convexity related to costs c, we prove that EVI flows are the limit of splitting schemes, providing assumptions for both implicit and explicit iterations.