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1. A potential-based construction of the increasing supermartingale coupling

   https://doi.org/10.1214/22-AAP1907  

   Bayraktar, Erhan; Deng, Shuoqing; Norgilas, Dominykas (October 2023, The Annals of Applied Probability)

   Friz, Peter (Ed.)

   The increasing supermartingale coupling, introduced by Nutz and Stebegg (Ann. Probab. 46 (2018) 3351–3398) is an extreme point of the set of “supermartingale” couplings between two real probability measures in convex-decreasing order. In the present paper we provide an explicit construction of a triple of functions, on the graph of which the increasing supermartingale coupling concentrates. In particular, we show that the increasing supermartingale coupling can be identified with the left-curtain martingale coupling and the antitone coupling to the left and to the right of a uniquely determined regime-switching point, respectively. Our construction is based on the concept of the shadow measure. We show how to determine the potential of the shadow measure associated to a supermartingale, extending the recent results of Beiglböck et al. (Electron. Commun. Probab. 27 (2022) 1–12) obtained in the martingale setting. 

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2. -ADIC -FUNCTIONS FOR UNITARY GROUPS

   https://doi.org/10.1017/fmp.2020.4  

   EISCHEN, ELLEN; HARRIS, MICHAEL; LI, JIANSHU; SKINNER, CHRISTOPHER (January 2020, Forum of Mathematics, Pi)

   This paper completes the construction of $$p$$ -adic $$L$$ -functions for unitary groups. More precisely, in Harris, Li and Skinner [‘ $$p$$ -adic $$L$$ -functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math. Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such $$p$$ -adic $$L$$ -functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and $$p$$ -adic differential operators [Eischen, ‘A $$p$$ -adic Eisenstein measure for unitary groups’, J. Reine Angew. Math. 699 (2015), 111–142; ‘ $$p$$ -adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble) 62 (1) (2012), 177–243], Part II of the present paper provides the calculations of local $$\unicode[STIX]{x1D701}$$ -integrals occurring in the Euler product (including at $$p$$ ). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method. 

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3. Tikhonov Regularization is Optimal Transport Robust under Martingale Constraints.

   Li, Jiajin; Lin, Sirui; Blanchet, Jose H.; Nguyen, Viet-Anh (July 2022, Advances in neural information processing systems)

   Distributionally robust optimization (DRO) has been shown to offer a principled way to regularize learning models. In this paper, we find that Tikhonov regularization is distributionally robust in an optimal transport sense (i.e. if an adversary chooses distributions in a suitable optimal transport neighborhood of the empirical measure), provided that suitable martingale constraints are also imposed. Further, we introduce a relaxation of the martingale constraints which not only provide a unified viewpoint to a class of existing robust methods but also lead to new regularization tools. To realize these novel tools, provably efficient computational algorithms are proposed. As a byproduct, the strong duality theorem proved in this paper can be potentially applied to other problems of independent interest. 

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4. The Brownian transport map

   https://doi.org/10.1007/s00440-024-01286-0  

   Mikulincer, Dan; Shenfeld, Yair (May 2024, Probability Theory and Related Fields)

   Abstract Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the Föllmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open. The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan–Lovász–Simonovits conjecture. We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors. 

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5. Quantitative Stability in the Geometry of Semi-discrete Optimal Transport

   https://doi.org/10.1093/imrn/rnaa355  

   Bansil, Mohit; Kitagawa, Jun (December 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We show quantitative stability results for the geometric “cells” arising in semi-discrete optimal transport problems. We first show stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. Next we show quantitative invertibility of the map taking dual variables to the measures of Laguerre cells, under a Poincarè-Wirtinger inequality. Combined with a regularity assumption equivalent to the Ma–Trudinger–Wang conditions of regularity in Monge-Ampère, this invertibility leads to stability of Laguerre cells in Hausdorff measure and also stability in the uniform norm of the dual potential functions, all stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Ampère regularity theory. 

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