Sala Seminari (Dip. Matematica).<\/p>\n
In optimal mass transport theory, many problems can be written in the Monge-Kantorovich form $$ \\inf\\{ \\int_{X\\times Y} c(x,y) \\, d\\gamma \\ :\\ \\gamma\\in \\Pi(\\mu,\\ u)\\}\\ ,\\eqno(1) $$ where $\\mu,\\ u$ are given probability measures on $X,Y$ and $c:X\\times Y \\to [0,+\\infty[$ is a cost function. Here the competitors are probability measures $\\gamma$ on $X\\times Y$ with marginals $\\mu$ and $\\ u$ respectively (transport plans). Let us recall that if an optimal transport plan $\\gamma \\in \\Pi(\\mu, \\ u)$ is carried by the graph of a map $T:X\\to Y$ i.e. if $$ = \\int_X \\varphi(x,Tx)\\, d\\mu \\quad,\\quad T^\\sharp \\mu= \\ u\\ ,$$ then $T$ solves the original Monge problem: \\ $ \\inf\\{ \\int_X c(x,Tx) \\, d\\mu\\ :\\ T^\\sharp \\mu= \\ u \\}.$ \\bigskip Here we are interested in a different case. Indeed in some applications to economy or in probability theory, it can be interesting to favour optimal plans which are non associated to a single valued transport map $T(x)$. The idea is then to consider, instead of $T(x)$, the family of conditional probabilities $\\gamma^x$ such that $$ = \\int_X (\\int_X \\varphi(x,y) d\\gamma^x(y))\\, d\\mu \\ ,$$ and to incorporate in problem $(1)$ an additional cost over $\\gamma^x$ as follows $$ \\inf \\left\\{ \\int_{X\\times X} c(x,y) \\, d\\gamma + \\int_X H(x, \\gamma^x) \\, d\\mu\\ :\\ \\gamma\\in \\Pi(\\mu,\\ u)\\right\\}\\ ,\\eqno(2) $$ being $H:(x,p) \\in X\\times \\mathcal{P}(X) \\to [0,+\\infty]$ a suitable non linear function. \\bigskip In this talk I will describe some results concerning problem $(2)$ (existence, duality principle, optimality conditions) and focus on specific examples where $X=Y$ and $X$ is a convex compact subset of $\\Rbb^d$. We will consider in particular the case where $H(x, p)= – \\text{var} (p)$ or where $H(x,\\cdot)$ is the indicator of a constraint on the barycenter of $p$ (martingale transport). \\medskip This is from a joined work with Thierry Champion and J.J. Albert. (https:\/\/www.cambridge.org\/core\/journals\/european-journal-of-applied-mathematics\/article\/new-class-of-costs-for-optimal-transport-planning\/E9DF63870C6B0524E6DB57B9FA3E4C03)<\/p>\n","protected":false},"excerpt":{"rendered":"
In optimal mass transport theory, many problems can be written in the Monge-Kantorovich form $$ \\inf\\{ \\int_{X\\times Y} c(x,y) \\, d\\gamma \\ :\\ \\gamma\\in \\Pi(\\mu,\\ u)\\}\\ ,\\eqno(1) $$ where $\\mu,\\ u$ are given probability measures on $X,Y$ and…<\/p>\n