Markov Kernels in Optimal Transport via Extending c-Cyclic Monotonicity

In this paper we show that we can use Markov kernels as a model for optimal transport. This new framework can be easily translated into the standard coupling formulation of optimal transport, and we show that we can use a coupling as a Markov kernel for many optimal transport problems. Using kernels allows us to extend optimal transport to signed measures and treats the support of the measure as the salient feature. This approach reveals additional structure for one-dimensional signed optimal transport.