PIMS Distinguished Speaker: Nassif Ghoussoub

Given a probability measure $\mu$ on a domain $\Omega \subset \R^d$, we consider the symmetric version of the Monge-Kantorovich theorem on the  product space $\Omega^N$, where the cost function $c$ is symmetric and all the prescribed marginals are equal to $\mu$. We show that the supremum is attained on a probability measure that is supported on the graph of functions of the form $x\to (x, Sx, S^2x,..., S^{N-1}x)$, where $S$ is a $\mu$-measure preserving transformation on $\Omega$ such that $S^N=I$ a.e. The result applies in particular  to the case where $c$ is given by the Newtonian potential associated to $N$ electrons. It then yields the existence of ``co-motion" functions that correspond to an interacting energy functional for ``strictly correlated electrons".

In another application, the result yields that for any integer $N$, essentially any non-degenerate vector field from $\Omega$ into $\R^d$ is $N$-cyclically monotone up to a measure preserving $N$-involution. This is joint work with Abbas Moameni.