Matthias Erbar, Martin Huesmann and Thomas Leblé The one-dimensional log-gas free energy has a unique minimiser arxiv e-print 1812.06929 2018 https://arxiv.org/abs/1812.06929
Michael Goldmann, Martin Huesmann and Felix Otto A large-scale regularity theory for the Monge-Ampere equation with rough data and application to the optimal matching problem arxiv e-print 1808.09250 2018 https://arxiv.org/abs/1808.09250
Martin Huesmann and Florian Stebegg Monotonicity preserving transformations of MOT and SEP Stochastic Process. Appl., 128(4): 1114--1134 2018 10.1016/j.spa.2017.07.005
2017
Mathias Beiglböck, Alexander M. G. Cox and Martin Huesmann he geometry of multi-marginal Skorokhod embedding arxiv e-print 1705.09505 2017 https://arxiv.org/abs/1705.09505
2016
Beatrice Acciaio, Alexander M. G. Cox and Martin Huesmann Model-independent pricing with insider information: A Skorokhod embedding approach arxiv e-print 1610.09124 2016 https://arxiv.org/abs/1610.09124
Mathias Beiglböck, Alexander M. G. Cox, Martin Huesmann, Nicolas Perkowski and David J. Prömel Pathwise super-replication via Vovk's outer measure ArXiv e-prints 2015 http://arxiv.org/abs/1504.03644
Matthias Erbar and Martin Huesmann Curvature bounds for configuration spaces Calculus of Variations and Partial Differential Equations, 54(1): 397-430 2015 http://dx.doi.org//10.1007/s00526-014-0790-1
Mathias Beiglböck, Alexander M.G. Cox and Martin Huesmann Optimal Transport and Skorokhod Embedding ArXiv eprints 2014 http://arxiv.org/abs/1307.3656
Abstract: The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability.
We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings. While previous constructions typically used particular properties of Brownian motion, our approach applies to all sufficiently regular Markov processes.
Fabio Cavalletti and Martin Huesmann Self-intersection of optimal geodesics Bulletin of the London Mathematical Society, 46(3): 653-656 2014 http://dx.doi.org/10.1112/blms/bdu026
Abstract: Let (X,d,m)(X,d,m) be a geodesic metric measure space. Consider a geodesic μtμt in the L2L2-Wasserstein space. Then as ss goes to tt, the support of μsμs and the support of μtμt have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. Given a geodesic of (X,d,m)(X,d,m) and t∈[0,1]t∈[0,1], we consider the set of times for which this geodesic belongs to the support of μtμt. We prove that tt is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying CD(K,∞)CD(K,∞). The non-branching property is not needed.
Abstract: Let (X,d,m) be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided (X,d,m) satisfies a new weak property concerning the behavior of m under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure contraction property.
We also prove a stability property for Assumption 1: If (X,d,m) satisfies Assumption 1 and View the MathML source, for some continuous function g>0, then also View the MathML source verifies Assumption 1. Since these changes in the reference measures do not preserve any Ricci type curvature bounds, this shows that our condition is strictly weaker than measure contraction property.
Martin Huesmann Optimal transport between random measures Annales de l’Institut Henri Poincaré (B) 2014 http://arxiv.org/abs/1206.3672
Abstract: We study couplings q∙ of two equivariant random measures λ∙ and μ∙ on a Riemannian manifold (M,d,m). Given a cost function we ask for minimizers of the mean transportation cost per volume. In case the minimal/optimal cost is finite and λω≪m we prove that there is a unique equivariant coupling minimizing the mean transportation cost per volume. Moreover, the optimal coupling is induced by a transportation map, q∙=(id,T)∗λ∙. We show that the optimal transportation map can be approximated by solutions to classical optimal transportation problems on bounded regions. In case of Lp−cost the optimal transportation cost per volume defines a metric on the space of equivariant random measure with unit intensity.
2013
Martin Huesmann and Karl-Theodor Sturm Optimal transport from Lebesgue to Poisson The Annals of Probability, 41(4): 2426-2478 2013 http://dx.doi.org/10.1214/12-AOP814
Abstract: This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite. Moreover, we give precise conditions for the latter which demonstrate a sharp threshold at d=2d=2. The cost will be defined in terms of an arbitrary increasing function of the distance.
The coupling will be realized by means of a transport map (“allocation map”) which assigns to each Poisson point a set (“cell”) of Lebesgue measure 1. In the case of quadratic costs, all these cells will be convex polytopes.