Peacock geodesics in Wasserstein space

Martingale optimal transport has attracted much attention due to its application in pricing and hedging in mathematical finance. The essential notion which makes martingale optimal transport different from optimal transport is peacock. A peacock is a sequence of measures which satisfies convex order property. In this paper we study peacock geodesics in Wasserstain space, and we hope this paper can provide some geometrical points of view to look at martingale optimal transport.     

Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, II

This is an addendum to the paper [K. Bacher, K.T. Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, J. Funct. Anal. 259 (2010) 28-56]. We prove the tensorization property for the curvature-dimension condition, add some detailed calculations - including explicit dependence of constants - and comment on assumptions and conjectures concerning the local-to-global statement in Bacher and Sturm (2010) [1] and Villani (2009) [6], respectively.     

Localization and tensorization properties of the curvature-dimension condition for metric measure spaces

This paper is devoted to the analysis of metric measure spaces satisfying locally the curvature-dimension condition CD(K,N) introduced by the second author and also studied by Lott & Villani. We prove that the local version of CD(K,N) is equivalent to a global condition CD^∗(K,N), slightly weaker than the (usual, global) curvature-dimension condition. This so-called reduced curvature-dimension condition CD^∗(K,N) has the local-to-global property. We also prove the tensorization property for CD^∗(K,N). As an application we conclude that the fundamental group π₁(M,x₀) of a metric measure space (M,d,m) is finite whenever it satisfies locally the curvature-dimension condition CD(K,N) with positive K and finite N.     

The Geodesic Problem in Quasimetric Spaces

In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality d(x,y)≤σ(d(x,z)+d(z,y)) for some constant σ≥1, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzelà theorem) still hold in quasimetric spaces. Moreover, we explore conditions under which a quasimetric will induce an intrinsic metric. As an example, we introduce a family of quasimetrics on the space of atomic probability measures. The associated intrinsic metrics induced by these quasimetrics coincide with the d _α metric studied early in the study of branching structures arisen in ramified optimal transportation. An optimal transport path between two atomic probability measures typically has a “tree shaped” branching structure. Here, we show that these optimal transport paths turn out to be geodesics in these intrinsic metric spaces. This work is supported by an NSF grant DMS-0710714.     

Stability of a 4th-order curvature condition arising in optimal transport theory

A certain curvature condition, introduced by Ma, Trudinger and Wang in relation with the regularity of optimal transport, is shown to be stable under Gromov-Hausdorff limits, even though the condition implicitly involves fourth order derivatives of the Riemannian metric. Two lines of reasoning are presented with slightly different assumptions, one purely geometric, and another one combining geometry and probability. Then a converse problem is studied: prove some partial regularity for the optimal transport on a perturbation of a Riemannian manifold satisfying a strong form of the Ma-Trudinger-Wang condition.     

Absolute continuity of Wasserstein geodesics in the Heisenberg group

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 end-points 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.     

Metric aspects of noncommutative homogeneous spaces

For a closed cocompact subgroup Γ of a locally compact group G, given a compact abelian subgroup K of G and a homomorphism satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deformations of the homogeneous space G/Γ, generalizing Rieffel's construction of quantum Heisenberg manifolds. We show that when G is a Lie group and G/Γ is connected, given any norm on the Lie algebra of G, the seminorm on induced by the derivation map of the canonical G-action defines a compact quantum metric. Furthermore, it is shown that this compact quantum metric space depends on ρ continuously, with respect to quantum Gromov-Hausdorff distances.     

距离空间上测度正则性的推广

本文给出了单调类与σ代数关系的充要条件,进一步得出距离空间的开集全体产生的最小单调类与产生的σ代数相等.利用这关系,证明了距离空间上的σ-有限测度也是正则测度.推广了距离空间上有限测度的正则性.     

Some Computational Aspects of Geodesic Convex Sets in a Simple Polygon

In this article, we deal with some computational aspects of geodesic convex sets. Motzkin-type theorem, Radon-type theorem, and Helly-type theorem for geodesic convex sets are shown. In particular, given a finite collection of geodesic convex sets in a simple polygon and an “oracle,” which accepts as input three sets of the collection and which gives as its output an intersection point or reports its nonexistence; we present an algorithm for finding an intersection point of this collection.     

Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time

The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles (“trees”). We describe a polynomial-time algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles.     

A note on the perturbations of compact quantum metric spaces

In this short note,we consider the perturbation of compact quantum metric spaces.We first show that for two compact quantum metric spaces(A,P) and(B,Q) for which A and B are subspaces of an order-unit space C and P and Q are Lip-norms on A and B respectively,the quantum Gromov–Hausdorff distance between(A,P) and(B,Q) is small under certain conditions.Then some other perturbation results on compact quantum metric spaces derived from spectral triples are also given.     

Fundamental properties of process distances

To quantify the difference of distinct stochastic processes it is not sufficient to consider the distance of their states and corresponding probabilities. Instead, the information, which evolves and accumulates over time and which is mathematically encoded by filtrations, has to be accounted for as well. The nested distance, also known as bicausal Wasserstein distance, recognizes this component and involves the filtration properly. This distance is of emerging importance due to its applications in stochastic analysis, stochastic programming, mathematical economics and other disciplines.This paper investigates the basic metric and topological properties of the nested distance on the space of discrete-time processes. In particular we prove that the nested distance generates a Polish topology, although the genuine space is not complete. Moreover we identify its completion to be the space of nested distributions, a space of generalized stochastic processes.     

Characterization of the Relations in Grzegorczyk's Hierarchy Revisited

In his 1953's paper, Grzegorczyk proved that a certain kind of relation classes of Grzegorczyk's hierarchy could be characterized inductively. We give a simpler version of this characterization.     

Consequences of Schanuel's condition for zeros of exponential terms

Assuming “Schanuel's Condition” for a certain class of exponential fields, Sturm's technique for polynomials in real closed fields can be extended to more complicated exponential terms in the corresponding exponential field. Hence for this class of terms the exact number of zeros can be calculated. These results give deeper insights into the model theory of exponential fields. MSC: 03C65, 03C60, 12L12.     

Sample Numbers and Optimal Lagrange Interpolation of Sobolev Spaces Wr1*

This paper investigates the optimal recovery of Sobolev spaces W^r₁[?1, 1], r ∈ N in the space L1[?1, 1]. They obtain the values of the sampling numbers of W^r₁[?1, 1] in L₁[?1, 1] and show that the Lagrange interpolation algorithms based on the extreme points of Chebyshev polynomials are optimal algorithms. Meanwhile, they prove that the extreme points of Chebyshev polynomials are optimal Lagrange interpolation nodes.     

On a general class of multi-valued weakly Picard mappings

The concept of weak contraction from the case of single-valued mappings is extended to multi-valued mappings and then corresponding convergence theorems for the Picard iteration associated to a multi-valued weak contraction are obtained. The main results in this paper extend, improve and unify a multitude of classical results in the fixed point theory of single and multi-valued contractive mappings and also improve recent results from the paper [P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl. 192 (1995), 655-666].     

Behavior of the constant in Korenblum's maximum principle

Let A^p (??) (p ≥ 1) be the Bergman space over the open unit disk ?? in the complex plane. Korenblum's maximum principle states that there is an absolute constant c ∈ (0, 1) (may depend on p), such that whenever |f (z)| ≤ |g (z)| (f, g ∈ A^p (??)) in the annulus c < |z | < 1, then ∥f ≤ ∥g ∥. For p ≥ 1, let c_p be the largest value of c for which Korenblum's maximum principle holds. In this note we prove that c_p → 1 as p → ∞. Thus we give a positive answer of a question of Hinkkanen. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)