The basic problem of optimal transportation consists in minimizing the expected costs E[c(X1, X2)] by varying the joint distribution (X1, X2) where the marginal distributions of the random variables X1 and X2 are fixed.Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that (Xi)i=1,2 is a martingale, that is,We establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific cost functions. In particular, we identify a martingale coupling that resembles the classic monotone quantile coupling in several respects. In analogy with the celebrated theorem of Brenier, the following behavior can be observed: If the initial distribution is continuous, then this "monotone martingale" is supported by the graphs of two functions T1, T2 : R → R.

We prove that no curvature-dimension bound CD(K, N) holds in any Heisenberg group Hn. On the contrary the measure contraction property M CP (0, 2n + 3) holds and is optimal for the dimension 2n + 3. For the non-existence of a curvature-dimension bound, we prove that the generalized "geodesic" Brunn-Minkowski inequality is false in Hn. We also show in a new and direct way, (and for all n ∈ N\{0}) that the general "multiplicative" Brunn-Minkowski inequality with dimension N > 2n + 1 is false.

Biological invasions offer interesting situations for observing how novel interactions between closely related, formerly allopatric species may trigger phenotypic evolution in situ. Assuming that successful invaders are usually filtered to be competitively dominant, invasive and native species may follow different trajectories. Natives may evolve traits that minimize the negative impact of competition, while trait shifts in invasives should mostly reflect expansion dynamics, through selection for colonization ability and transiently enhanced mutation load at the colonization front. These ideas were tested through a large-scale common-garden experiment measuring life-history traits in two closely related snail species, one invasive and one native, co-occurring in a network of freshwater ponds in Guadeloupe. We looked for evidence of recent evolution by comparing uninvaded or recently invaded sites with long-invaded ones. The native species adopted a life history favoring rapid population growth (i.e., increased fecundity, earlier reproduction, and increased juvenile survival) that may increase its prospects of coexistence with the more competitive invader. We discuss why these effects are more likely to result from genetic change than from maternal effects. The invader exhibited slightly decreased overall performances in recently colonized sites, consistent with a moderate expansion load resulting from local founder effects. Our study highlights a rare example of rapid life-history evolution following invasion.