Existence of an optimal-coupling Markov kernel for lower semi-continuous transport costs

Determine whether, for a complete separable metric space S, a Markov operator P on S, and a lower semi-continuous semi-distance φ: S×S→[0,∞), there exists a Markov transition kernel Q on S² such that for all x,y∈S, Q((x,y),·)∈Π(δ_x P, δ_y P) attains the Kantorovich cost and satisfies D_φ(δ_x P, δ_y P)=∫ φ(u,v) Q((x,y), d(u,v)).

Background

The paper proves Lemma \ref{lemma:ced}, which characterizes the Dobrushin contraction coefficient via Dirac measures. In the proof, the authors construct continuous approximations (φk) of a lower semi-continuous cost φ to obtain measurability of the mapping (x,y)↦Dφ(δ_x P, δ_y P).

They note that when φ is continuous, Corollary 5.22 in Villani (2008) guarantees the existence of a Markov transition Q on S² such that, for every x,y∈S, Q((x,y),·) is an optimal transport plan between δx P and δ_y P, yielding the representation Dφ(δ_x P, δ_y P)=∫ φ(u,v) Q((x,y), d(u,v)).

However, under the weaker assumption that φ is only lower semi-continuous, the existence of such a measurable selection Q (a Markov kernel that assigns to each (x,y) an optimal coupling achieving the Kantorovich cost) is not established, prompting the explicit open question.

References

Assuming only that $\phi$ is l.s.c., eq:D_lim implies that the mapping $(x,y)\mapsto D_{\phi}(\delta_{x}P,\delta_{y}P) $ is measurable but it is unclear whether or not there exists a Markov kernel $Q$ such that pced3 holds.

pced3:

Dϕ(δxP,δyP)= ∫ ϕ(u,v) Q((x,y),d(u,v)),∀x,y∈S.D_{\phi}(\delta_{x}P,\delta_{y}P)=~\int~\phi(u,v)~Q((x,y),d(u,v)),\quad\forall x,y\in S.

— On the Kantorovich contraction of Markov semigroups (2511.08111 - Moral et al., 11 Nov 2025) in Remark after the proof of Lemma \ref{lemma:ced}, Section 4.1