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Optimal transport between random measures
Abstract
We study couplings of two equivariant random measures
and 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 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, We show that the optimal
transportation map can be approximated by solutions to classical optimal
transportation problems on bounded regions. In case of cost the optimal
transportation cost per volume defines a metric on the space of equivariant
random measure with unit intensity.
... In [5,4] it was shown that there is a unique optimal coupling between the Lebesgue measure λ d on R d and an invariant random measure µ on R d of unit intensity provided that the asymptotic mean transportation cost ...
... The main results of [5,4] show that there is a unique optimal coupling between λ and µ provided that c ∞ < ∞. In particular, eventhough there are arbitrarily many asymptotically optimal couplings there is a unique invariant one. ...
... Denote by q n the optimal semicoupling between λ and ½ [0,n) µ. By Proposition 4.2 in [4], there is a transport map T n and a density ρ n such that q n = (id, T n ) * (ρ n λ). Put l n := inf{x : ρ n (x) > 0} and r n := sup{x : ρ n (x) > 0}. ...
We show that there is a sharp threshold in dimension one for the transport cost between the Lebesgue measure and an invariant random measure of unit intensity to be finite. We show that for \emph{any} such random measure the cost are infinite provided that the first central moments diverge. Furthermore, we establish simple and sharp criteria, based on the variance of , for the cost to be finite for .
... C). To this end, we return to the cost function (1.1) which has been analysed for jointly invariant random measures with fixed joint distribution by Sturm and the second author in [HS13,Hue16]. For two jointly invariant random measures ξ, η defined on the same probability space (Ω, F , P) we say that a random measure q : (Ω, F , P) → M(R d × R d ) is an invariant/equivariant coupling of ξ and η iff for all A ∈ B(R d ) ...
... We want to stress that the existence of an optimal invariant q attaining c(ξ, η) a.s. is the most difficult part of the statement. This follows using techniques developed in [HS13,Hue16] once an optimal distribution coupling Q is fixed. To see why (1.3) holds, observe that for any q ∈ cpl e (ξ, η) we have by invariance ...
... so that the difference between the two sides in (1.3) is essentially whether the lim sup is inside or outside the expectation (plus the choice of an invariant q). The main tool to show equality is the ergodic theorem together with approximation techniques for the optimal q from [HS13,Hue16]. ...
We develop a theory of optimal transport for stationary random measures with a particular focus on stationary point processes. This provides us with a notion of geodesic distance between distributions of stationary random measures and induces a natural displacement interpolation between them. In the setting of stationary point processes we leverage this transport distance to give a geometric interpretation for the evolution of infinite particle systems with stationary distribution. Namely, we characterise the evolution of infinitely many Brownian motions as the gradient flow of the specific relative entropy w.r.t.~the Poisson process. Further, we establish displacement convexity of the specific relative entropy along optimal interpolations of point processes.
... The proof of Theorem 1.2 is based on the optimal transport techniques for random measures, introduced in [HS13] and [Hue16]. Let us sketch the argument. ...
... [HP05,CPPR10,LMT14,HS13]. Allocations and equivariant couplings or transports between two general random measures ξ and η have been investigated e.g. in [LT09,LMT14,LTT18,Hue16]. We also refer to [AT93,Tho96] for the origin of shift-couplings and to [LT09] and [LT21, Remark 2.2] for results on shift couplings resulting from allocations between general random measures. ...
... ; see Definition 5.3 in [Hue16]. Local optimality of optimal semicouplings can be shown as in Proposition 3.1 and Theorem 3.6 in [HS13]. ...
Let be a pair of jointly stationary, ergodic random measures of equal finite intensity. A balancing allocation is a translation-invariant (equivariant) map such that the image measure of under T is . We show that as soon as does not charge small sets, i.e.\ does not give mass to -rectifiable sets, there is always a balancing allocation T which is measurably depending only on , i.e. T is a factor.
... In [5,4] it was shown that there is a unique optimal coupling between the Lebesgue measure λ d on R d and an invariant random measure µ on R d of unit intensity provided that the asymptotic mean transportation cost ...
... The main results of [5,4] show that there is a unique optimal coupling between λ and µ provided that c ∞ < ∞. In particular, eventhough there are arbitrarily many asymptotically optimal couplings there is a unique invariant one. ...
... Denote by q n the optimal semicoupling between λ and ½ [0,n) µ. By Proposition 4.2 in [4], there is a transport map T n and a density ρ n such that q n = (id, T n ) * (ρ n λ). Put l n := inf{x : ρ n (x) > 0} and r n := sup{x : ρ n (x) > 0}. ...
We show that there is a sharp threshold in dimension one for the transport
cost between the Lebesgue measure and an invariant random measure
of unit intensity to be finite. We show that for \emph{any} such random
measure the cost are infinite provided that the first central moments
diverge. Furthermore, we establish simple and
sharp criteria, based on the variance of , for the cost to be
finite for .
... The proof of Theorem 1.2 is based on the optimal transport techniques for random measures, introduced in Huesmann and Sturm (2013) and Huesmann (2016). Let us sketch the argument. ...
... This work together with Holroyd and Peres (2005) Huesmann and Sturm (2013). Allocations and equivariant couplings or transports between two general random measures ξ and η have been investigated e.g. in Last and Thorisson (2009);Last et al. (2014Last et al. ( , 2018; Huesmann (2016). We also refer to Aldous and Thorisson (1993); Thorisson (1996) for the origin of shift-couplings and to Last and Thorisson (2009) and Last and Thorisson (2023, Remark 2.2) for results on shift couplings resulting from allocations between general random measures. ...
... Transports of random measures and point processes have been studied on more general phase spaces. For further background we refer to [20,12,3,4,10,9,5]. The existence of an extra head was implicit in an abstract group result in [20], but in that paper there was no hint at an explicit pathwise method of finding an extra head. ...
... The mass of the source prefers to be allocated as close as possible. The paper [5] pursues a different approach, based on the minimization of expected transport costs (defined in the Palm sense). It is shown that if the expected transport cost is finite and the source is absolutely continuous, then there exists a unique optimal allocation that can be locally approximated with solutions to the classical Monge problem (see [22]). ...
We consider two jointly stationary and ergodic random measures and on the real line with equal intensities. An allocation is an equivariant random mapping from to . We give sufficient and partially necessary conditions for the existence of allocations transporting to . An important ingredient of our approach is to introduce a transport kernel balancing and , provided these random measures are mutually singular. In the second part of the paper, we apply this result to the path decomposition of a two-sided Brownian motion into three independent pieces: a time reversed Brownian motion on , an excursion distributed according to a conditional It\^o's law and a Brownian motion starting after this excursion. An analogous result holds for Bismut's excursion law.
... Nx +U,x+V = EN U,V almost surely. Our interest in this problem is motivated on the one hand by work on geometric properties of matchings by Holroyd [13] and Holroyd et al. [15,14], and on the other hand by work on optimally coupling random measures by the first author and Sturm [17] and the first author [16]. In [14], Holroyd, Janson, and Wästlund analyze (stationary) matchings satisfying the local optimality condition (1.2) with the exponent 2 replaced by γ ∈ [−∞, ∞]. ...
... The cases γ ≥ 1 were left open, but see [15] and [13] for several open questions for d = 2 and in particular γ = 1. On the other hand, the first author and Sturm [17,16] develop an optimal transport approach to this (and related) problems . They identify the point sets {X}, {Y } with the counting measures (1.3) and seek a stationary coupling Q between µ and ν minimizing the cost ...
We show that there is no cyclically monotone stationary matching of two independent Poisson processes in dimension d=2. The proof combines the harmonic approximation result from \cite{GHO} with local asymptotics for the two-dimensional matching problem for which we give a new self-contained proof using martingale arguments.
... In all these contributions, the transport distances are essentially distances between point processes having a density w.r.t. a reference point process, they are "classical" transport distances. Building on [HS13,Hue16] and motivated by the results of [EHL21], in the recent article [EHJM24] a transport distance C p per volume for stationary point processes was introduced. It was shown that the gradient flow of the specific entropy w.r.t. ...
We construct a non-local Benamou-Brenier-type transport distance on the space of stationary point processes and analyse the induced geometry. We show that our metric is a specific variant of the transport distance recently constructed in [DSHS24]. As a consequence, we show that the Ornstein-Uhlenbeck semigroup is the gradient flow of the specific relative entropy w.r.t. the newly constructed distance. Furthermore, we show the existence of stationary geodesics, establish 1-geodesic convexity of the specific relative entropy, and derive stationary analogues of functional inequalities such as a specific HWI inequality and a specific Talagrand inequality. One of the key technical contributions is the existence of solutions to the non-local continuity equation between arbitrary point processes.
... This work assumes that Φ assigns zero measure to every (d − 1)-dimensional Lipschitz manifold, which is stronger than the assumptions of Theorem 1.2 (it is claim in [7] that the condition is sharp, but Theorem 1.2 shows that this is not the case). The method of the proof is by using optimal transport and an extension of Monge's theorem to stationary random measures provided in [6]. So this work does not prove Theorem 1.1 on the existence of factor point processes. ...
In this article, we show that every stationary random measure on R^d that is essentially free (i.e., has no symmetries a.s.) admits a point process as a factor. As a result, we improve the results of Last and Thorisson (2022) on the existence of a factor balancing allocation between ergodic pairs of stationary random measures Φ and Ψ with equal intensities. In particular, we prove that such an allocation exists if Φ is diffuse and either (Φ, Ψ) is essentially free or Φ assigns zero measure to every (d − 1)-dimensional affine hyperplane. The main result is deduced from an existing result in descriptive set theory, that is, the existence of lacunary sections. We also weaken the assumption of being essentially free to the case where a discrete group of symmetries is allowed.
... We have chosen here to treat each of those cases separately because treating discrete measures as in Section 5 is more explicit then this alternative method. Huesmann (2016), unique optimal allocations between jointly stationary random measures on geodesic manifolds were constructed, assuming the (Palm) average cost to be finite and the source to be absolutely continuous. Stable transports between general (jointly stationary) random measures ξ and η on R d were constructed and studied in Haji-Mirsadeghi and Khezeli (2016). ...
... The construction in [7] involved a Gale-Shapley algorithm resulting in a 'stable' allocation while the construction in [2] used a gravitational force field to obtain an 'economical' allocation. In [8], unique optimal allocations between jointly stationary random measures on geodesic manifolds were constructed, assuming the (Palm) average cost to be finite and the source to be absolutely continuous. Stable transports between general (jointly stationary) random measures ξ and η on R d were constructed and studied in [5]. ...
We consider two jointly stationary and ergodic random measures and on with equal finite intensities, assuming to be diffuse. An allocation is a random mapping taking to in a translation invariant way. We construct allocations transporting the diffuse to arbitrary , under the mild condition of existence of an `auxiliary' point process which is needed only in the case when is diffuse. When that condition does not hold we show by a counterexample that an allocation transporting to need not exist.
... We cannot argue via displacement convexity directly on the level of P 0 ; P 1 since they are probability measures on infinite point configurations. Optimal transport theory for random stationary measures as initiated in [7,10,11] is not yet developed well enough to be directly applicable. Instead, we use transport theory between finite measures together with a careful approximation argument relying on screening of electric fields. ...
We prove that, at every positive temperature, the infinite‐volume free energy of the one‐dimensional log‐gas, or beta‐ensemble, has a unique minimizer, which is the Sine‐beta process arising from random matrix theory. We rely on a quantitative displacement convexity argument at the level of point processes, and on the screening procedure introduced by Sandier‐Serfaty.
... We cannot argue via displacement convexity directly on the level of P 0 , P 1 since they are probability measures on infinite point configurations. Optimal transport theory for random stationary measures as initiated in [HS13,Hue16,EH15] is not yet developed well enough to be directly applicable. Instead, we use transport theory between finite measures together with a careful approximation argument relying on screening of electric fields. ...
We prove that, at every positive temperature, the infinite-volume free energy of the one dimensional log-gas, or beta-ensemble, has a unique minimiser, which is the Sine-beta process arising from random matrix theory. We rely on a quantitative displacement convexity argument at the level of point processes, and on the screening procedure introduced by Sandier-Serfaty.
We show that there is no cyclically monotone stationary matching of two independent Poisson processes in dimension d=2d=2. The proof combines the harmonic approximation result from Goldman et al. (Commun. Pure Appl. Math. 74:2483–2560, 2021) with local asymptotics for the two-dimensional matching problem for which we give a new self-contained proof using martingale arguments.
We consider two jointly stationary and ergodic random measures and on the real line with equal intensities. An allocation is an equivariant random mapping from to . We give sufficient and partially necessary conditions for the existence of allocations transporting to . An important ingredient of our approach is to introduce a transport kernel balancing and , provided these random measures are mutually singular. In the second part of the paper, we apply this result to the path decomposition of a two-sided Brownian motion into three independent pieces: a time reversed Brownian motion on , an excursion distributed according to a conditional It\^{o}'s law and a Brownian motion starting after this excursion. An analogous result holds for Bismut's excursion law.
We introduce a robust and feature-capturing surface reconstruction and simplification method that turns an input point set into a low triangle-count simplicial complex. Our approach starts with a (possibly non-manifold) simplicial complex filtered from a 3D Delaunay triangulation of the input points. This initial approximation is iteratively simplified based on an error metric that measures, through optimal transport, the distance between the input points and the current simplicial complex—both seen as mass distributions. Our approach is shown to exhibit both robustness to noise and outliers, as well as preservation of sharp features and boundaries. Our new feature-sensitive metric between point sets and triangle meshes can also be used as a post-processing tool that, from the smooth output of a reconstruction method, recovers sharp features and boundaries present in the initial point set.
We prove existence and uniqueness of optimal maps on spaces
under the assumption that the starting measure is absolutely continuous. We
also discuss how this result naturally leads to the notion of exponentiation.
We have been investigating the formation and evolution of clusters of
galaxies from low redshift to high redshift using the NIR/MIR images
taken with the AKARI/IRC, as part of the AKARI mission program: CLEVL.
We present an infrared view of these clusters. It is found that the
cluster galaxies detected in MIR are divided into two groups: MIR-blue
galaxies with (N3-S11)<0 and MIR-red galaxies with (N3-S11)>0 .
MIR-red galaxies are star forming galaxies with varying star formation
rates, corresponding to late type galaxies. MIR-blue galaxies are red
sequence galaxies with varying stellar ages or with low-level star
formation activity. The fraction of the MIR-red galaxies varies
significantly from cluster to cluster.
The potential function of the optimal transportation problem satisfies a partial differential equation of Monge-Ampre type. In this paper we introduce the notion of a generalized solution, and prove the existence and uniqueness of generalized solutions of the problem. We also prove the solution is smooth under certain structural conditions on the cost function.
Givenn random red points on the unit square, the transportation cost between them is tipically √n logn.
We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation
problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [24], [30] for a priori estimates of the corresponding Monge–Ampère equation. It is expressed by a socalled cost-sectional curvature being non-negative. We show that when the cost function is the squared distance of a Riemannian manifold, the cost-sectional
curvature yields the sectional curvature. As a consequence, if the manifold does not have non-negative sectional curvature
everywhere, the optimal transport map cannot be continuous for arbitrary smooth positive data. The non-negativity of the cost-sectional curvature is shown to be equivalent to the connectedness
of the contact set between any cost-convex function (the proper generalization of a convex function) and any of its supporting
functions. When the cost-sectional curvature is uniformly positive, we obtain that optimal maps are continuous or Hölder continuous
under quite weak assumptions on the data, compared to what is needed in the Euclidean case. This case includes the quadratic
cost on the round sphere.
Let (M,g) be a connected compact manifold, C3 smooth and without boundary, equipped with a Riemannian distance d(x,y). If s : M ® M s : M \to M is merely Borel and never maps positive volume into zero volume, we show s = t °u s = t \circ u factors uniquely a.e. into the composition of a map t(x) = expx[-Ñy(x)] t(x) = {\rm exp}_x[-\nabla\psi(x)] and a volume-preserving map u : M ® M u : M \to M , where
y: M ® \bold R \psi : M \to {\bold R} satisfies the additional property that (yc)c = y (\psi^c)^c = \psi with yc(y) :=inf{c(x,y) - y(x)|x Î M} \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} and c(x,y) = d2(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.The results are obtained by solving a Riemannian version of the Monge--Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution f ³ 0 f \ge 0 of mass in L1(M) onto another. Parallel results for other strictly convex cost functions c(x,y) ³ 0 c(x,y) \ge 0 of the Riemannian distance on non-compact manifolds are briefly discussed.
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=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.
A systematic study of random Laguerre tessellations, weighted
generalisations of the well-known Voronoi tessellations, is
presented. We prove that every normal tessellation with convex
cells in dimension three and higher is a Laguerre tessellation.
Tessellations generated by stationary marked Poisson processes are
then studied in detail. For these tessellations, we obtain
integral formulae for geometric characteristics and densities of
the typical k-faces. We present a formula for the linear
contact distribution function and prove various limit results for
convergence of Laguerre to Poisson-Voronoi tessellations. The
obtained integral formulae are subsequently evaluated numerically
for the planar case, demonstrating their applicability for
practical purposes.
Thesis (Ph. D.)--Princeton University, 1994. Includes bibliographical references (leaves 152-155).
We construct the entropic measure on compact manifolds of any dimension. It is defined as the push forward of the Dirichlet process (a random probability measure, well-known to exist on spaces of any dimension) under the conjugation map
This conjugation map is a continuous involution. It can be regarded as the canonical extension to higher-dimensional spaces of a map between probability measures on 1-dimensional spaces characterized by the fact that the distribution functions of μ and C(μ) are inverse to each other.
We also present a heuristic interpretation of the entropic measure as
d \mathbb{P}^\beta(\mu) = \frac{1}{\rm Z} {\rm exp} (- \beta \cdot {\rm Ent}(\mu | m)) \cdot d \mathbb{P}^0(\mu).
A complete characterization of multivariate random variables with minimum L2 Wasserstein-distance is proved by means of duality theory and convex analysis. This characterization allows to determine explicitly the optimal couplings for several multivariate distributions. A partial solution of this problem has been found in recent papers by Knott and Smith.
. Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of innite components, the expected degree, and the topology of the components. Our fundamental tool is a new mass-transport technique that has been occasionally used elsewhere and is developed further here. Perhaps surprisingly, these investigations of group-invariant percolation produce results that are new in the Bernoulli setting. Most notably, we prove that critical Bernoulli percolation on any nonamenable Cayley graph has no innite clusters. More generally, the same is true for any nonamenable graph with a unimodular transitive automorphism group. We show that G is amenable i for all < 1, there is a G-invariant site percolation process ! on X with P[x 2 !] > for all vertices x and with no innite components. When G is not amenable, a threshold < 1 ...
For d>=3, we construct a non-randomized, fair and translation-equivariant allocation of Lebesgue measure to the points of a standard Poisson point process in R^d, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical in the sense that the "allocation diameter", defined as the diameter X of the basin of attraction containing the origin, is a random variable with a rapidly decaying tail. Specifically, we have the tail bound: P(X > R) < C exp[ -c R(log R)^(alpha_d) ], for all R>2, where: alpha_d = (d-2)/d for d>=4; alpha_3 can be taken as any number <-4/3; and C,c are positive constants that depend on d and alpha_d. This is the first construction of an allocation rule of Lebesgue measure to a Poisson point process with subpolynomial decay of the tail P(X>R).
We consider the zeroes of the random Gaussian entire function ( , . . . are Gaussian i.i.d. complex random variables) and show that their basins under the gradient flow of the random potential partition the complex plane into domains of equal area.
We find three characteristic exponents 1, 8/5, and 4 of this random partition: the probability that the diameter of a particular basin is greater than R is exponentially small in R; the probability that a given point z lies at a distance larger than R from the zero, it is attracted to decays as ; and the probability that, after throwing away 1% of the area of the basin, its diameter is still larger than R decays as .
We also introduce a combinatorial procedure that modifies a small portion of each basin in such a way that the probability that the diameter of a particular modified basin is greater than R decays as .
Let be a discrete set in . Call the elements of centers. The well-known Voronoi tessellation partitions into polyhedral regions (of varying sizes) by allocating each site of to the closest center. Here we study ``fair'' allocations of to in which the regions allocated to different centers have equal volumes. We prove that if is obtained from a translation-invariant point process, then there is a unique fair allocation which is stable in the sense of the Gale--Shapley marriage problem. (I.e., sites and centers both prefer to be allocated as close as possible, and an allocation is said to be unstable if some site and center both prefer each other over their current allocations.) We show that the region allocated to each center is a union of finitely many bounded connected sets. However, in the case of a Poisson process, an infinite volume of sites are allocated to centers further away than . We prove power law lower bounds on the allocation distance of a typical site. It is an open problem to prove any upper bound in .
Let \Pi be an ergodic simple point process on R^d and let \Pi^* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of \Pi and \Pi^*; that is, one can select a (random) point Y of \Pi such that translating \Pi by -Y yields a configuration whose law is that of \Pi^*. We construct shift couplings in which Y and \Pi^* are functions of \Pi, and prove that there is no shift coupling in which \Pi is a function of \Pi^*. The key ingredient is a deterministic translation-invariant rule to allocate sets of equal volume (forming a partition of R^d) to the points of \Pi. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let \Gamma be an ergodic random element of {0,1}^{Z^d} and let \Gamma^* be \Gamma conditioned on \Gamma(0)=1. A shift coupling X of \Gamma and \Gamma^* is called an extra head scheme. We show that there exists an extra head scheme which is a function of \Gamma if and only if the marginal E[\Gamma(0)] is the reciprocal of an integer. When the law of \Gamma is product measure and d\geq3, we prove that there exists an extra head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].
Let M and \bar M be n -dimensional manifolds equipped with suitable Borel probability measures ρ and \bar ρ . For subdomains M and \bar M of ℝ^n , Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c \in C^4 (M × M) to guarantee smoothness of the optimal map pushing ρ forward to \bar ρ ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost c induces on the product space M × \bar M . We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry.
Using the pseudo-Riemannian structure, we extend Ma, Trudinger and Wang’s conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principle characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and \bar M of ℝ^n . This maximum principle plays a key role in Loeper’s Hölder continuity theory of optimal o maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered.
In this section we introduce and discuss some basic properties of percolation, a fundamental random process on graphs. For background on percolation see [Gri99].
Although point processes are just integer-valued random measures, their importance justifies a separate treatment, and their special features yield to techniques not readily applicable to general random measures. The first and last parts of the chapter summarize results for point processes, which parallel those for random measures—existence theorems, moment structure, and generating functional—as well as furnishing illustrative (and important) examples. Many of the results are special cases of the corresponding results in Chapter 6, while others are extensions from the context of finite point processes in Chapter 5. The remaining part of the chapter, on the avoidance functions and intensity measures, deals with properties that are peculiar to point processes and for which the extensions to general random measures are not easily found.
The following paper is reproduced from a Russian journal of the character of our own Proceedings of the National Academy of Sciences, Comptes Rendus (Doklady) de I'Académie des Sciences de I'URSS, 1942, Volume XXXVII, No. 7–8. The author is one of the most distinguished of Russian mathematicians. He has made very important contributions in pure mathematics in the theory of functional analysis, and has made equally important contributions to applied mathematics in numerical analysis and the theory and practice of computation. Although his exposition in this paper is quite terse and couched in mathematical language which may be difficult for some readers of Management Science to follow, it is thought that this presentation will: (1) make available to American readers generally an important work in the field of linear programming, (2) provide an indication of the type of analytic work which has been done and is being done in connection with rational planning in Russia, (3) through the specific examples mentioned indicate the types of interpretation which the Russians have made of the abstract mathematics (for example, the potential and field interpretations adduced in this country recently by W. Prager were anticipated in this paper).
It is to be noted, however, that the problem of determining an effective method of actually acquiring the solution to a specific problem is not solved in this paper. In the category of development of such methods we seem to be, currently, ahead of the Russians.—A. Charnes, Northwestern Technological Institute and The Transportation Center.
This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.
We prove existence of optimal maps in non branching spaces with Ricci curvature bounded from below. The approach we adopt makes no use of Kantorovich potentials.
We prove that no curvature-dimension bound CD(K, N) holds in any Heisenberg group H(n). On the contrary, the measure contraction property MC P(0, 2n + 3) holds and is optimal for the dimension 2n + 3. For the nonexistence of a curvature-dimension bound, we prove that the generalized "geodesic" Brunn-Minkowski inequality is false in H(n). We also show in a new and direct way (and for all n is an element of N\{0}), that the general "multiplicative" Brunn-Minkowski inequality with dimension N > 2n + 1 is false.
We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound
and dim(M) ⩽ N.
The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact.
Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers
theorem. Moreover, it implies the doubling property and local, scale-invariant Poincar inequalities on balls. In particular,
it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.
We find conditions under which two measure preserving actions of two groups
on the same space have a common fundamental domain. Our results apply to
commuting actions with separate fundamental domains, lattices in groups of
polynomial growth, and some semidirect products. We prove that two lattices of
equal co-volume in a group of polynomial growth, one acting on the left, the
other on the right, have a common fundamental domain.
Introduction The Kantorovich duality Geometry of optimal transportation Brenier's polar factorization theorem The Monge-Ampere equation Displacement interpolation and displacement convexity Geometric and Gaussian inequalities The metric side of optimal transportation A differential point of view on optimal transportation Entropy production and transportation inequalities Problems Bibliography Table of short statements Index.
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 \tilde m = g \cdot m , for some continuous function g > 0 , then also (X,d,\tilde m) 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.
Given a probability space (X, μ) and a bounded domain Ω in ℝd equipped with the Lebesgue measure |·| (normalized so that |Ω| = 1), it is shown (under additional technical assumptions on X and Ω) that for every vector-valued function u ∈ Lp (X, μ; ℝd) there is a unique “polar factorization” u = ∇Ψs, where Ψ is a convex function defined on Ω and s is a measure-preserving mapping from (X, μ) into (Ω, |·|), provided that u is nondegenerate, in the sense that μ(u−1(E)) = 0 for each Lebesgue negligible subset E of ℝd.
Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real-valued functions are unified.
The Monge-Ampère equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge-Kantorovich” problem.
1 Some elementary examples
2 Optimal transport plans: existence and regularity
3 The one dimensional case
4 The ODE version of the optimal transport problem
5 The PDE version of the optimal transport problem and the p-laplacian approximation
6 Existence of optimal transport maps
7 Regularity and uniqueness of the transport density
8 The Bouchitt-Buttazzo mass optimization problem
9 Appendix: some measure theoretic results
References
Given two densities f and g, we consider the problem of transporting a fraction m Î [0,min{||f||L1,||g||L1}]{m \in [0,\min\{\|f\|_{L^1},\|g\|_{L^1}\}]} of the mass of f onto g minimizing a transportation cost. If the cost per unit of mass is given by |x − y|2, we will see that uniqueness of solutions holds for m Î [||fÙg||L1,min{||f||L1,||g||L1}]{m \in [\|f\wedge g\|_{L^1},\min\{\|f\|_{L^1},\|g\|_{L^1}\}]} . This extends the result of Caffarelli and McCann in Ann Math (in print), where the authors consider two densities with disjoint supports. The free boundaries of the active
regions are shown to be (n − 1)-rectifiable (provided the supports of f and g have Lipschitz boundaries), and under some weak regularity assumptions on the geometry of the supports they are also locally
semiconvex. Moreover, assuming f and g supported on two bounded strictly convex sets
W,L Ì \mathbb Rn{{\Omega,\Lambda \subset \mathbb {R}^n}} , and bounded away from zero and infinity on their respective supports, C0,aloc{C^{0,\alpha}_{\rm loc}} regularity of the optimal transport map and local C
1 regularity of the free boundaries away from WÇL{{\Omega\cap \Lambda}} are shown. Finally, the optimal transport map extends to a global homeomorphism between the active regions.
The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov's curvature bounded below, Monge's transport problem for the quadratic cost admits a unique solution.
We consider the Monge–Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value ∞. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions provided that {c=∞} is the union of a closed set and a negligible set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish that transport plans are strongly c-monotone if and only if they satisfy a “better” notion of optimality called robust optimality.
We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected O(nde(d ln(n+1))1/4) ...
This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
The author introduces a measure contraction property on a metric space X equipped with a measure . Roughly speaking, this means that for every set with and for every , the normalized restriction of the measure µ to A can be transported in a controlled way along geodesics to the Dirac measure . The measure contraction property can be regarded as a generalization of the lower curvature bound; in particular it is shown that Aleksandrov spaces with a lower curvature bound satisfy the measure contraction property, and that for Riemannian manifolds the measure contraction property is equivalent to the lower bound of the Ricci curvature.
Several consequences of the measure contraction property are proved in the paper, e.g. an estimate on the Hausdorff dimension and diameter of X (a generalization of the Bonnet-Myers theorem) and the Bishop-Gromov comparison theorem for the volume of balls in X.
Finally, it is shown that the measure contraction property is stable under Gromov-Hausdorff convergence and that a family of compact metric spaces with the same total measure and the same measure contraction property is compact in the Gromov-Hausdorff topology.
Reviewed by Jana Björn
Let be a two-sided standard Brownian motion. An
\emph{unbiased shift} of B is a random time T, which is a measurable
function of B, such that is a Brownian motion
independent of . We characterise unbiased shifts in terms of allocation
rules balancing additive functionals of B. For any probability distribution
on we construct a stopping time with the above properties
such that has distribution . In particular, we show that if we
travel in time according to the clock of local time we always see a two-sided
Brownian motion. A crucial ingredient of our approach is a new theorem on the
existence of allocation rules balancing jointly stationary diffuse random
measures on . We also study moment and minimality properties of unbiased
shifts.
Given a strictly increasing, continuous function ,
based on the cost functional , we
define the -Wasserstein distance between
probability measures on some metric space (X,d). The function
will be assumed to admit a representation
as a composition of a convex and a concave function and , resp.
Besides convex functions and concave functions this includes all
functions.
For such functions we extend the concept of Orlicz spaces,
defining the metric space of measurable functions such that, for instance,
The allocation problem for a Poisson point process is to find a way to
partition the space to parts of equal size, and assign the parts to the
configuration points in a measurable, "deterministic" (equivariant) way. The
goal is to make the diameter of the part assigned to a configuration point have
fast decay. We present an algorithm for , that achieves an tail, which is optimal up to c. This improves the best previously
known allocation rule, the gravitational allocation, which has tail.
Suppose that red and blue points occur as independent Poisson processes of equal intensity in R^d, and that the red points are matched to the blue points via straight edges in a translation-invariant way. We address several closely related properties of such matchings. We prove that there exist matchings that locally minimize total edge length in d=1 and d>=3, but not in the strip R x [0,1]. We prove that there exist matchings in which every bounded set intersects only finitely many edges in d>=2, but not in d=1 or in the strip. It is unknown whether there exists a matching with no crossings in d=2, but we prove positive answers to various relaxations of this question. Several open problems are presented. Comment: 21 pages
We introduce and study invariant (weighted) transport-kernels balancing stationary random measures on a locally compact Abelian group. The first main result is an associated fundamental invariance property of Palm measures, derived from a generalization of Neveu's exchange formula. The second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transport-kernels. We then introduce (in a nonstationary setting) the concept of mass-stationarity with respect to a random measure, formalizing the intuitive idea that the origin is a typical location in the mass. The third main result of the paper is that a measure is a Palm measure if and only if it is mass-stationary. Comment: Published in at http://dx.doi.org/10.1214/08-AOP420 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
The theory of Palm measures is developed for the general context of stationary random measures on locally compact second countable groups (not necessarily Abelian). Connections are made with transport kernels, allocation problems and shift coupling.
In this paper the regularity of optimal transportation potentials defined on
round spheres is investigated. Specifically, this research generalises the
calculations done by Loeper, where he showed that the strong (A3) condition of
Trudinger and Wang is satisfied on the round sphere, when the cost-function is
the geodesic distance squared. In order to generalise Loeper's calculation to a
broader class of cost-functions, the (A3) condition is reformulated via a
stereographic projection that maps charts of the sphere into Euclidean space.
This reformulation subsequently allows one to verify the (A3) condition for any
case where the cost-fuction of the associated optimal transportation problem
can be expressed as a function of the geodesic distance between points on a
round sphere. With this, several examples of such cost-functions are then
analysed to see whether or not they satisfy this (A3) condition.