Optimal transportation of processes with infinite Kantorovich distance. Independence and symmetry

Understand

We consider probability measures on $\mathbb{R}^{\infty}$ and study optimal transportation mappings for the case of infinite Kantorovich distance.

• Our examples include 1) quasi-product measures, 2) measures with certain symmetric properties, in particular, exchangeable and stationary measures.
• We show in the latter case that existence problem for optimal transportation is closely related to ergodicity of the target measure.
• In particular, we prove existence of the symmetric optimal transportation for a certain class of stationary Gibbs measures.