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Martin Huesmann

Publications and source records attributed to Martin Huesmann.

At least 19 recordsLinked to original sources
arXiv2025

Gap metrics for stationary point processes and quantitative convexity of the free energy

In this article, we are interested in convexity properties of the free energy for stationary point processes on $\mathbb R$ w.r.t.\ a new geometry inspired by optimal transport. We will show for a rich class of pairwise interaction energies A) quantified strict convexity of the free energy implying uniqueness of minimizers B) existence of a gradient flow curve of the free energy w.r.t. the new metric converging exponentially fast to the unique minimizer. Examples for energies for which A holds include logarithmic or Riesz interactions with parameter $0<s<1$, examples for which A and B hold are hypersingular Riesz or Yukawa interactions.

math.PR↗
arXiv2025

Non-local Wasserstein Geometry, Gradient Flows, and Functional Inequalities for Stationary Point Processes

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.

math.PR↗
arXiv2024

Asymptotics for Random Quadratic Transportation Costs

We establish the validity of asymptotic limits for the general transportation problem between random i.i.d. points and their common distribution, with respect to the squared Euclidean distance cost, in any dimension larger than three. Previous results were essentially limited to the two (or one) dimensional case, or to distributions whose absolutely continuous part is uniform. The proof relies upon recent advances in the stability theory of optimal transportation, combined with functional analytic techniques and some ideas from quantitative stochastic homogenization. The key tool we develop is a quantitative upper bound for the usual quadratic optimal transportation problem in terms of its boundary variant, where points can be freely transported along the boundary. The methods we use are applicable to more general random measures, including occupation measure of Brownian paths, and may open the door to further progress on challenging problems at the interface of analysis, probability, and discrete mathematics.

math.PR↗
arXiv2024

(Non)-hyperuniformity of perturbed lattices

We ask whether a stationary lattice in dimension $d$ whose points are shifted by identically distributed but possibly dependent perturbations remains hyperuniform. When $d = 1$ or $2$, we show that it is the case when the perturbations have a finite $d$-moment, and that this condition is sharp. When $d \geq 3$, we construct arbitrarily small perturbations such that the resulting point process is not hyperuniform. As a side remark of independent interest, we exhibit hyperuniform processes with arbitrarily slow decay of their number variance.

math.PR↗
arXiv2024

The link between hyperuniformity, Coulomb energy, and Wasserstein distance to Lebesgue for two-dimensional point processes

We investigate the interplay between three possible properties of stationary point processes: i) Finite Coulomb energy with short-scale regularization, ii) Finite $2$-Wasserstein transportation distance to the Lebesgue measure and iii) Hyperuniformity. In dimension $2$, we prove that i) implies ii), which is known to imply iii), and we provide simple counter-examples to both converse implications. However, we prove that ii) implies i) for processes with a uniformly bounded density of points, and that i) - finiteness of the regularized Coulomb energy - is equivalent to a certain property of quantitative hyperuniformity that is just slightly stronger than hyperuniformity itself. Our proof relies on the classical link between $H^{-1}$-norm and $2$-Wasserstein distance between measures, on the screening construction for Coulomb gases (of which we present an adaptation to $2$-Wasserstein space which might be of independent interest), and on recent necessary and sufficient conditions for the existence of stationary "electric" fields compatible with a given stationary point process.

math.PR↗
arXiv2023

There is no stationary $p$-cyclically monotone Poisson matching in 2D

We show that for $p>1$ there is no $p$-cyclically monotone stationary matching of two independent Poisson processes in dimension $d=2$. The proof combines the $p$-harmonic approximation result from \cite[Theorem 1.1]{koch23} with local asymptotics for the two-dimensional matching problem. Moreover, we prove a.s. local upper bounds of the correct order in the case $p>1$, which, to the best of our knowledge, are not readily available in the current literature.

math.PR↗
arXiv2023

Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy

We develop a theory of optimal transport for stationary random measures with a focus on stationary point processes and construct a family of distances on the set of stationary random measures. These induce a natural notion of interpolation between two stationary random measures along a shortest curve connecting 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 point process. Further, we establish displacement convexity of the specific relative entropy along optimal interpolations of point processes and establish an stationary analogue of the HWI inequality, relating specific entropy, transport distance, and a specific relative Fisher information.

math.PR↗
arXiv2023

Transportation of random measures not charging small sets

Let $(\xi,\eta)$ be a pair of jointly stationary, ergodic random measures of equal finite intensity. A balancing allocation is a translation-invariant (equivariant) map $T:\mathbb{R}^d\to\mathbb{R}^d$ such that the image measure of $\xi$ under $T$ is $\eta$. We show that as soon as $\xi$ does not charge small sets, i.e.\ does not give mass to $(d-1)$-rectifiable sets, there is always a balancing allocation $T$ which is measurably depending only on $(\xi,\eta)$, i.e. $T$ is a factor.

math.PR↗
arXiv2022

Wasserstein Asymptotics for the Empirical Measure of Fractional Brownian Motion on a Flat Torus

We establish asymptotic upper and lower bounds for the Wasserstein distance of any order $p\ge 1$ between the empirical measure of a fractional Brownian motion on a flat torus and the uniform Lebesgue measure. Our inequalities reveal an interesting interaction between the Hurst index $H$ and the dimension $d$ of the state space, with a "phase-transition" in the rates when $d=2+1/H$, akin to the Ajtai-Koml\'os-Tusn\'ady theorem for the optimal matching of i.i.d. points in two-dimensions. Our proof couples PDE's and probabilistic techniques, and also yields a similar result for discrete-time approximations of the process, as well as a lower bound for the same problem on $\mathbb{R}^d$.

math.PR↗
arXiv2021

There is no stationary cyclically monotone Poisson matching in 2d

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.

math.PR↗
arXiv2021

A fluctuation result for the displacement in the optimal matching problem

The aim of this paper is to justify in dimensions two and three the ansatz of Caracciolo et al. stating that the displacement in the optimal matching problem is essentially given by the solution to the linearized equation i.e. the Poisson equation. Moreover, we prove that at all mesoscopic scales, this displacement is close in suitable negative Sobolev spaces to a curl-free Gaussian free field. For this we combine a quantitative estimate on the difference between the displacement and the linearized object, which is based on the large-scale regularity theory recently developed in collaboration with F. Otto, together with a qualitative convergence result for the linearized problem.

math.PR↗
arXiv2021

Quantitative linearization results for the Monge-Ampère equation

This paper is about quantitative linearization results for the Monge-Ampère equation with rough data. We develop a large-scale regularity theory and prove that if a measure $μ$ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson equation. The main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures. This is used in a Campanato iteration which transfers the information through the scales.

math.AP↗
arXiv2021

Shadows and Barriers

We show an intimate connection between solutions of the Skorokhod Embedding Problem which are given as the first hitting time of a barrier and the concept of shadows in martingale optimal transport. More precisely, we show that a solution $τ$ to the Skorokhod Embedding Problem between $μ$ and $ν$ is of the form $τ= \inf \{t \geq 0 : (X_t,B_t) \in \mathcal{R}\}$ for some increasing process $(X_t)_{t \geq 0}$ and a barrier $\mathcal{R}$ if and only if there exists a time-change $(T_l)_{l \geq 0}$ such that for all $l \geq 0$ the equation $$\mathbb{P}[B_τ \in \cdot , τ\geq T_l] = \mathcal{S}^ν(\mathbb{P}[B_{T_l} \in \cdot , τ\geq T_l])$$ is satisfied, i.e.\ the distribution of $B_τ$ on the event that the Brownian motion is stopped after $T_l$ is the shadow of the distribution of $B_{T_l}$ on this event in the terminal distribution $ν$. This equivalence allows us to construct new families of barrier solutions that naturally interpolate between two given barrier solutions. We exemplify this by an interpolation between the Root embedding and the left-monotone embedding.

math.PR↗
arXiv2020

Model-independent pricing with insider information: a Skorokhod embedding approach

In this paper, we consider the pricing and hedging of a financial derivative for an insider trader, in a model-independent setting. In particular, we suppose that the insider wants to act in a way which is independent of any modelling assumptions, but that she observes market information in the form of the prices of vanilla call options on the asset. We also assume that both the insider's information, which takes the form of a set of impossible paths, and the payoff of the derivative are time-invariant. This setup allows us to adapt recent work of Beiglboeck, Cox and Huesmann (2016) to prove duality results and a monotonicity principle, which enables us to determine geometric properties of the optimal models. Moreover, we show that this setup is powerful, in that we are able to find analytic and numerical solutions to certain pricing and hedging problems.

q-fin.MF↗
arXiv2020

Shadow martingales -- a stochastic mass transport approach to the peacock problem

Given a family of real probability measures $(\mu_t)_{t\geq 0}$ increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in \cite{BeJu16,NuStTa17}. As input data we take an increasing family of measures $(\nu^\alpha)_{\alpha \in [0,1]}$ with $\nu^\alpha(\mathbb{R})=\alpha$ that are submeasures of $\mu _0$, called a parametrization of $\mu_0$. Then, for any $\alpha$ we define an evolution $(\eta^\alpha_t)_{t\geq 0}$ of the measure $\nu^\alpha=\eta^\alpha_0$ across our peacock by setting $\eta^\alpha_t$ equal to the obstructed shadow of $\nu^\alpha$ in $(\mu _s)_{s \in [0,t]}$. We identify conditions on the parametrization $(\nu^\alpha)_{\alpha\in [0,1]}$ such that this construction leads to a unique martingale measure $\pi$, the shadow martingale, without any assumptions on the peacock. In the case of the left-curtain parametrization $(\nu_{\text{lc}}^\alpha)_{\alpha \in [0,1]}$ we identify the shadow martingale as the unique solution to a continuous-time version of the martingale optimal transport problem. Furthermore, our method enriches the knowledge on the Predictable Representation Property (PRP) since any shadow martingale comes with a canonical Choquet representation in extremal Markov martingales.

math.PR↗
arXiv2019

Martingale Benamou--Brenier: a probabilistic perspective

In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. We suggest a Benamou-Brenier type formulation of the martingale transport problem for given $d$-dimensional distributions $μ, ν$ in convex order. The unique solution $M^*=(M_t^*)_{t\in [0,1]}$ of this problem turns out to be a Markov-martingale which has several notable properties: In a specific sense it mimics the movement of a Brownian particle as closely as possible subject to the conditions $M^*_0\simμ, M^*_1\sim ν$. Similar to McCann's displacement-interpolation, $M^*$ provides a time-consistent interpolation between $μ$ and $ν$. For particular choices of the initial and terminal law, $M^*$ recovers archetypical martingales such as Brownian motion, geometric Brownian motion, and the Bass martingale. Furthermore, it yields a natural approximation to the local vol model and a new approach to Kellerer's theorem. This article is parallel to the work of Huesmann-Trevisan, who consider a related class of problems from a PDE-oriented perspective.

math.PR↗