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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
$ <\gamma, \varphi(x,y)> = \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,\nu)\right\}\ ,\quad (2) $
being $H:(x,p) \in X\times \mathcal{P}(X) \to [0,+\infty]$ a suitable non linear function.
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).
This is from a joined work with Thierry Champion and J.J. Alibert.
http://cvgmt.sns.it/seminar/683/
When | Wed Apr 3, 2019 4pm – 5pm Coordinated Universal Time |
Where | Sala Seminari (Dipartimento di Matematica di Pisa) (map) |