Transfunctions and their connections to Plans, Markov Operators and Optimal Transport

TL;DR

This paper explores the mathematical properties of transfunctions, their connections to Markov operators and plans, and develops approximation methods relevant to optimal transport problems.

Contribution

It characterizes transfunctions related to Markov operators and plans, introduces their adjoints, and develops approximation techniques for weakly-continuous transfunctions.

Findings

Characterized transfunctions corresponding to Markov operators and plans.

Defined and analyzed the adjoint of transfunctions in different settings.

Developed approximation methods for transfunctions, applicable to optimal transport.

Abstract

A transfunction is a function which maps between sets of finite measures on measurable spaces. In this paper we characterize transfunctions that correspond to Markov operators and to plans; such a transfunction will contain the "instructions" common to several Markov operators and plans. We also define the adjoint of transfunctions in two settings and provide conditions for existence of adjoints. Finally, we develop approximations of identity in each setting and use them to approximate weakly-continuous transfunctions with simple transfunctions; one of these results can be applied to some optimal transport problems to approximate the optimal cost with simple Markov transfunctions.