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.

European food-deceptive orchids generally flower early in spring and rely on naı¨ve pollinators for their reproduction. Some species however, flower later in the summer, when many other rewarding plants species are also in bloom. In dense flowering communities, deceptive orchids may suffer from competition for pollinator resources, or might alternatively benefit from higher community attractiveness. We investigated the pollination strategy of the deceptive species Traunsteinera globosa, and more specifically whether it benefited from the presence of coflowering rewarding species. We carried out a population survey to quantify the density and reproductive success of the orchid as well as the density of all coflowering species. Our results suggest that the deceptive orchid not only benefited from the presence of coflowering species, but that interestingly the density of the species Trifolium pratense was significantly positively correlated with the orchid's reproductive success. This species might simply act as a magnet species attracting pollinators near T. globosa, or could influence the orchid reproductive fitness through a more species-specific interaction. We propose that morphological or colour similarities between the two species should be investigated in more detail to decipher this pollination facilitation effect.

Abstract. The (left-)curtain coupling, introduced by Beiglböck and the author is an extreme element of the set of "martingale" couplings between two real probability measures in convex order. It enjoys remarkable properties with respect to order relations and a minimisation problem inspired by the theory of optimal transport. An explicit representation and a number of further noteworthy attributes have recently been established by Henry-Labordère and Touzi. In the present paper we prove that the curtain coupling depends continuously on the prescribed marginals and quantify this with Lipschitz estimates. Moreover, we investigate the Markov composition of curtain couplings as a way of associating Markovian martingales with peacocks.

We give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in Gozlan, Roberto, Samson and Tetali (J. Funct. Anal. 273 (2017) 3327–3405). Optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling. We also establish some connections with Caffarelli's contraction theorem (Caffarelli, Comm. Math. Phys. 214 (2000) 547–563).

Plant reproductive success within a patch may depend on plant aggregation through pollinator attraction. For rewardless plants that lack rewards for pollinators, reproductive success may rely strongly on the learning abilities of pollinators. These abilities depend on relative co-flowering rewarding and rewardless plant species spatial distributions. We investigated the effect of aggregation on the reproductive success of a rewardless orchid by setting up 16 arrays in a factorial design with two levels of intraspecific aggregation for both a rewardless orchid and a rewarding co-flowering species. Our results show that increasing aggregation of both species negatively influenced the reproductive success of the rewardless plants. To our knowledge, this is the first experimental study demonstrating negative effects of aggregation on reproductive success of a rewardless species due both to its own spatial aggregation and that of a co-flowering rewarding species. We argue that pollinator learning behaviour is the key driver behind this result.

In this paper we answer to a question raised by Ambrosio and Rigot [L. Ambrosio, S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 208 (2) (2004) 261-301] proving that any interior point of a Wasserstein geodesic in the Heisenberg group is absolutely continuous if one of the endpoints is. Since our proof relies on the validity of the so-called Measure Contraction Property and on the fact that the optimal transport map exists and the Wasserstein geodesic is unique, the absolute continuity of Wasserstein geodesic also holds for Alexandrov spaces with curvature bounded from below.