Consider three absolutely continuous probability measures $\mu$, $u$, and $\nu$ on $\mathbb R^d$ ($d \geq 1$), all of which have finite second moments. A transport map from $\mu$ to $\nu$ is called optimal if it is the solution to the following constrained optimization problem:
$$ \min\int \|\boldsymbol x - T(\boldsymbol x)\|_2^2 ~d \mu(\boldsymbol x) \quad \text{subject to } T_\#\mu = \nu $$
where $T_\#\mu = \nu$ is the push forward operation meaning that if $\boldsymbol X \sim \mu$, then $T(\boldsymbol X) \sim \nu$. Also, denote $T^{\mu\nu}$ as the optimal transport map for convenience. The existence and $\mu$-a.e uniqueness is well guaranteed under stated assumptions.
My question: do we have some kind of transferability for optimal transport maps? That is, is $T^{\mu u} \circ T^{u \nu} = T^{\mu \nu}$ $\mu$-a.e.? If not, can someone provide a valid counterexample? Many thanks.
Some side notes:
- The statement is true when $d = 1$, it can be proved using the monotonicity of CDFs and Brenier-McCann theorem together;
- The result is not true if we don't assume absolute continuity. The counterexample should be easy to construct, but I am more interested in the case where the stated assumptions (bold) hold.