@article{21018, abstract = {In this paper, we review recent results on stability and instability in logarithmic Sobolev inequalities, with a particular emphasis on strong norms. We consider several versions of these inequalities on the Euclidean space, for the Lebesgue and the Gaussian measures, and discuss their differences in terms of moments and stability. We give new and direct proofs, as well as examples and discuss the stability of a logarithmic uncertainty principle. Although we do not cover all aspects of the topic, we hope to contribute to establishing the state of the art.}, author = {Brigati, Giovanni and Dolbeault, Jean and Simonov, Nikita}, issn = {2730-9657}, journal = {La Matematica}, publisher = {Springer Nature}, title = {{Logarithmic Sobolev Inequalities: A review on stability and instability results}}, doi = {10.1007/s44007-025-00180-y}, volume = {5}, year = {2026}, } @article{21132, abstract = {We unify the variational hypocoercivity framework established by D. Albritton, S. Armstrong, J.-C. Mourrat, and M. Novack [2], with the notion of second-order lifts of reversible diffusion processes, recently introduced by A. Eberle and the second author [30]. We give an abstract, yet fully constructive, presentation of the theory, so that it can be applied to a large class of linear kinetic equations. As this hypocoercivity technique does not twist the reference norm, we can recover accurate and sharp convergence rates in various models. Among those, adaptive Langevin dynamics (ALD) is discussed in full detail and we show that for near-quadratic potentials, with suitable choices of parameters, it is a near-optimal second-order lift of the overdamped Langevin dynamics. As a further consequence, we observe that the Generalised Langevin Equation (GLE) is also a second-order lift, as the standard (kinetic) Langevin dynamics are, of the overdamped Langevin dynamics. Then, convergence of (GLE) cannot exceed ballistic speed, i.e. the square root of the rate of the overdamped regime. We illustrate this phenomenon with explicit computations in a benchmark Gaussian case.}, author = {Brigati, Giovanni and Lörler, Francis and Wang, Lihan}, issn = {1937-5077}, journal = {Kinetic and Related Models}, pages = {34--55}, publisher = {American Institute of Mathematical Sciences}, title = {{Hypocoercivity meets lifts}}, doi = {10.3934/krm.2025020}, volume = {20}, year = {2026}, } @article{21504, abstract = {Selecting an appropriate divergence measure is a critical aspect of machine learning, as it directly impacts model performance. Among the most widely used, we find the Kullback–Leibler (KL) divergence, originally introduced in kinetic theory as a measure of relative entropy between probability distributions. Just as in machine learning, the ability to quantify the proximity of probability distributions plays a central role in kinetic theory. In this paper, we present a comparative review of divergence measures rooted in kinetic theory, highlighting their theoretical foundations and exploring their potential applications in machine learning and artificial intelligence.}, author = {Auricchio, Gennaro and Brigati, Giovanni and Giudici, Paolo and Toscani, Giuseppe}, issn = {1793-6314}, journal = {Mathematical Models and Methods in Applied Sciences}, publisher = {World Scientific Publishing}, title = {{From kinetic theory to AI: A rediscovery of high-dimensional divergences and their properties}}, doi = {10.1142/S0218202526410010}, year = {2026}, } @article{20865, abstract = {We prove the convergence of a modified Jordan–Kinderlehrer–Otto scheme to a solution to the Fokker–Planck equation in Ω e R^d with general—strictly positive and temporally constant—Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum. In the special case where Ω is an interval in R1, we prove that such a solution is a gradient flow—curve of maximal slope—within a suitable space of measures, endowed with a modified Wasserstein distance. Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107–130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41–88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary ∂Ω throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure Ω . The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when Ω is an interval in R1, we find a formula for the descending slope of this geodesically nonconvex functional.}, author = {Quattrocchi, Filippo}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {1}, publisher = {Springer Nature}, title = {{Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions}}, doi = {10.1007/s00526-025-03193-1}, volume = {65}, year = {2026}, } @article{18632, abstract = {For an arbitrary dimension (Formula presented.), we study: the polyharmonic Gaussian field (Formula presented.) on the discrete torus (Formula presented.), that is the random field whose law on (Formula presented.) given by (Formula presented.) where (Formula presented.) is the Lebesgue measure and (Formula presented.) is the discrete Laplacian; the associated discrete Liouville quantum gravity (LQG) measure associated with it, that is, the random measure on (Formula presented.) (Formula presented.) where (Formula presented.) is a regularity parameter. As (Formula presented.), we prove convergence of the fields (Formula presented.) to the polyharmonic Gaussian field (Formula presented.) on the continuous torus (Formula presented.), as well as convergence of the random measures (Formula presented.) to the LQG measure (Formula presented.) on (Formula presented.), for all (Formula presented.). }, author = {Dello Schiavo, Lorenzo and Herry, Ronan and Kopfer, Eva and Sturm, Karl Theodor}, issn = {1522-2616}, journal = {Mathematische Nachrichten}, number = {1}, pages = {244--281}, publisher = {Wiley}, title = {{Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous}}, doi = {10.1002/mana.202400169}, volume = {298}, year = {2025}, } @article{20040, abstract = {Contractive coupling rates have been recently introduced by Conforti as a tool to establish convex Sobolev inequalities (including modified log-Sobolev and Poincaré inequality) for some classes of Markov chains. In this work, for most of the examples discussed by Conforti, we use contractive coupling rates to prove stronger inequalities, in the form of curvature lower bounds (in entropic and discrete Bakry–Émery sense) and geodesic convexity of some entropic functionals. In addition, we recall and give straightforward generalizations of some notions of coarse Ricci curvature, and we discuss some of their properties and relations with the concepts of couplings and coupling rates: as an application, we show exponential contraction of the p-Wasserstein distance for the heat flow in the aforementioned examples.}, author = {Pedrotti, Francesco}, issn = {1050-5164}, journal = {The Annals of Applied Probability}, number = {1}, pages = {196 -- 250}, publisher = {Institute of Mathematical Statistics}, title = {{Contractive coupling rates and curvature lower bounds for Markov chains}}, doi = {10.1214/24-aap2113}, volume = {35}, year = {2025}, } @article{20050, abstract = {We prove upper bounds on the L∞-Wasserstein distance from optimal transport between strongly log-concave probability densities and log-Lipschitz perturbations. In the simplest setting, such a bound amounts to a transport-information inequality involving the L∞-Wasserstein metric and the relative L∞-Fisher information. We show that this inequality can be sharpened significantly in situations where the involved densities are anisotropic. Our proof is based on probabilistic techniques using Langevin dynamics. As an application of these results, we obtain sharp exponential rates of convergence in Fisher’s infinitesimal model from quantitative genetics, generalising recent results by Calvez, Poyato, and Santambrogio in dimension 1 to arbitrary dimensions.}, author = {Khudiakova, Kseniia and Maas, Jan and Pedrotti, Francesco}, issn = {1050-5164}, journal = {The Annals of Applied Probability}, number = {3}, pages = {1913--1940}, publisher = {Institute of Mathematical Statistics}, title = {{L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model}}, doi = {10.1214/25-aap2162}, volume = {35}, year = {2025}, } @article{20155, abstract = {We study time averages for the norm of solutions to kinetic Fokker–Planck equations associated with general Hamiltonians. We provide fully explicit and constructive decay estimates for systems subject to a confining potential, allowing fat-tail, subexponential and (super-)exponential local equilibria, which also include the classic Maxwellian case. The key step in our estimates is a modified Poincaré inequality, obtained via a Lions–Poincaré inequality and an averaging lemma.}, author = {Brigati, Giovanni and Stoltz, Gabriel}, issn = {1095-7154}, journal = {SIAM Journal on Mathematical Analysis}, number = {4}, pages = {3587--3622}, publisher = {Society for Industrial and Applied Mathematics}, title = {{How to construct explicit decay rates for kinetic Fokker–Planck equations?}}, doi = {10.1137/24M1700351}, volume = {57}, year = {2025}, } @article{20591, abstract = {In this paper we derive estimates for the Hessian of the logarithm (log-Hessian) for solutions to the heat equation. For initial data in the form of log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian admits an explicit, uniform (in space) lower bound. This yields a new estimate for the Lipschitz constant of a transport map pushing forward the standard Gaussian to a measure in this class. On the other hand, we show that assuming only fast decay of the tails of the initial datum does not suffice to guarantee uniform log-Hessian upper bounds.}, author = {Brigati, Giovanni and Pedrotti, Francesco}, issn = {1083-589X}, journal = {Electronic Communications in Probability}, publisher = {Institute of Mathematical Statistics}, title = {{Heat flow, log-concavity, and Lipschitz transport maps}}, doi = {10.1214/25-ECP717}, volume = {30}, year = {2025}, } @article{20814, abstract = {We characterize all semigroups sandwiched between the semigroup of a Dirichlet form and the semigroup of its active main part. In case the Dirichlet form is regular, we give a more explicit description of the quadratic forms of the sandwiched semigroups in terms of pairs consisting of an open set and a measure on an abstract boundary.}, author = {Keller, Matthias and Lenz, Daniel and Schmidt, Marcel and Schwarz, Michael and Wirth, Melchior}, issn = {1572-929X}, journal = {Potential Analysis}, publisher = {Springer Nature}, title = {{Boundary representations of intermediate forms between a regular Dirichlet form and its active main part}}, doi = {10.1007/s11118-025-10251-y}, volume = {64}, year = {2025}, } @article{19565, abstract = {Measuring distances in a multidimensional setting is a challenging problem, which appears in many fields of science and engineering. In this paper, to measure the distance between two multivariate distributions, we introduce a new measure of discrepancy which is scale invariant and which, in the case of two independent copies of the same distribution, and after normalization, coincides with the scaling invariant multidimensional version of the Gini index recently proposed in [P. Giudici, E. Raffinetti and G. Toscani, Measuring multidimensional inequality: A new proposal based on the Fourier transform, preprint (2024), arXiv:2401.14012 ]. A byproduct of the analysis is an easy-to-handle discrepancy metric, obtained by application of the theory to a pair of Gaussian multidimensional densities. The obtained metric does improve the standard metrics, based on the mean squared error, as it is scale invariant. The importance of this theoretical finding is illustrated by means of a real problem that concerns measuring the importance of Environmental, Social and Governance factors for the growth of small and medium enterprises. }, author = {Auricchio, Gennaro and Brigati, Giovanni and Giudici, Paolo and Toscani, Giuseppe}, issn = {1793-6314}, journal = {Mathematical Models and Methods in Applied Sciences}, number = {5}, pages = {1267--1296}, publisher = {World Scientific Publishing}, title = {{Multivariate Gini-type discrepancies}}, doi = {10.1142/s0218202525500174}, volume = {35}, year = {2025}, } @article{19625, abstract = {We introduce operator-valued twisted Araki–Woods algebras. These are operator-valued versions of a class of second quantization algebras that includes q-Gaussian and q-Araki–Woods algebras and also generalize Shlyakhtenko’s von Neumann algebras generated by operator-valued semicircular variables. We develop a disintegration theory that reduces the isomorphism type of operator-valued twisted Araki–Woods algebras over type I factors to the scalar-valued case. Moreover, these algebras come with a natural weight, and we characterize its modular theory. We also give sufficient criteria that guarantee the factoriality of these algebras.}, author = {Kumar, R. Rahul and Wirth, Melchior}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, number = {5}, publisher = {Springer Nature}, title = {{Operator-valued twisted Araki–Woods algebras}}, doi = {10.1007/s00220-025-05285-7}, volume = {406}, year = {2025}, } @article{21322, abstract = {Habitat fragmentation poses a significant risk to population survival, causing both demographic stochasticity and genetic drift within local populations to increase, thereby increasing genetic load. Higher load causes population numbers to decline, which reduces the efficiency of selection and further increases load, resulting in a positive feedback that may drive entire populations to extinction. Here, we investigate this eco-evolutionary feedback in a metapopulation consisting of local demes connected via migration, with individuals subject to deleterious mutation at a large number of loci. We first analyze the determinants of load under soft selection, where population sizes are fixed, and then build on this to understand hard selection, where population sizes and load coevolve. We show that under soft selection, very little gene flow (less than one migrant per generation) is enough to prevent fixation of deleterious alleles. By contrast, much higher levels of migration are required to mitigate load and prevent extinction when selection is hard, with critical migration thresholds for metapopulation persistence increasing sharply as the genome-wide deleterious mutation rate becomes comparable to the baseline population growth rate. Moreover, critical migration thresholds are highest if deleterious mutations have intermediate selection coefficients but lower if alleles are predominantly recessive rather than additive (due to more efficient purging of recessive load within local populations). Our analysis is based on a combination of analytical approximations and simulations, allowing for a more comprehensive understanding of the factors influencing load and extinction in fragmented populations.}, author = {Olusanya, Oluwafunmilola O and Khudiakova, Kseniia and Sachdeva, Himani}, issn = {1537-5323}, journal = {The American Naturalist}, number = {6}, pages = {617--636}, publisher = {University of Chicago Press}, title = {{Genetic load, eco-evolutionary feedback, and extinction in metapopulations}}, doi = {10.1086/735562}, volume = {205}, year = {2025}, } @phdthesis{20563, abstract = {The theory of optimal transport provides an elegant and powerful description of many evolution equations as gradient flows. The primary objective of this thesis is to adapt and extend the theory to deal with important equations that are not covered by the classical framework, specifically boundary value problems and kinetic equations. Additionally, we establish new results in periodic homogenization for discrete dynamical optimal transport and in quantization of measures. Section 1.1 serves as an invitation to the classical theory of optimal transport, including the main definitions and a selection of well-established theorems. Sections 1.2-1.5 introduce the main results of this thesis, outline the motivations, and review the current state of the art. In Chapter 2, we consider the Fokker–Planck equation on a bounded set with positive Dirichlet boundary conditions. We construct a time-discrete scheme involving a modification of the Wasserstein distance and, under weak assumptions, prove its convergence to a solution of this boundary value problem. In dimension 1, we show that this solution is a gradient flow in a suitable space of measures. Chapter 3 presents joint work with Giovanni Brigati and Jan Maas. We introduce a new theory of optimal transport to describe and study particle systems at the mesoscopic scale. We prove adapted versions of some fundamental theorems, including the Benamou–Brenier formula and the identification of absolutely continuous curves of measures. Chapter 4 presents joint work with Lorenzo Portinale. We prove convergence of dynamical transportation functionals on periodic graphs in the large-scale limit when the cost functional is asymptotically linear. Additionally, we show that discrete 1-Wasserstein distances converge to 1-Wasserstein distances constructed from crystalline norms on R d . Chapter 5 concerns optimal empirical quantization: the problem of approximating a measure by the sum of n equally weighted Dirac deltas, so as to minimize the error in the p-Wasserstein distance. Our main result is an analog of Zador’s theorem, providing asymptotic bounds for the minimal error as n tends to infinity. }, author = {Quattrocchi, Filippo}, issn = {2663-337X}, keywords = {optimal transport, kinetic equations, boundary value problems, quantization, gradient flows, homogenization}, pages = {240}, publisher = {Institute of Science and Technology Austria}, title = {{Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures}}, doi = {10.15479/AT-ISTA-20563}, year = {2025}, } @unpublished{20569, abstract = {This is the first part of a general description in terms of mass transport for time-evolving interacting particles systems, at a mesoscopic level. Beyond kinetic theory, our framework naturally applies in biology, computer vision, and engineering. The central object of our study is a new discrepancy d between two probability distributions in position and velocity states, which is reminiscent of the 2-Wasserstein distance, but of second-order nature. We construct d in two steps. First, we optimise over transport plans. The cost function is given by the minimal acceleration between two coupled states on a fixed time horizon T. Second, we further optimise over the time horizon T > 0. We prove the existence of optimal transport plans and maps, and study two time-continuous characterisations of d. One is given in terms of dynamical transport plans. The other one -- in the spirit of the Benamou--Brenier formula -- is formulated as the minimisation of an action of the acceleration field, constrained by Vlasov's equations. Equivalence of static and dynamical formulations of d holds true. While part of this result can be derived from recent, parallel developments in optimal control between measures, we give an original proof relying on two new ingredients: Galilean regularisation of Vlasov's equations and a kinetic Monge--Mather shortening principle. Finally, we establish a first-order differential calculus in the geometry induced by d, and identify solutions to Vlasov's equations with curves of measures satisfying a certain d-absolute continuity condition. One consequence is an explicit formula for the d-derivative of such curves.}, author = {Brigati, Giovanni and Maas, Jan and Quattrocchi, Filippo}, booktitle = {arXiv}, keywords = {optimal transport, kinetic theory, second-order discrepancy, Vlasov equation, Wasserstein splines.}, title = {{Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures}}, doi = {10.48550/arXiv.2502.15665}, year = {2025}, } @article{18158, abstract = {We study the geometry of Poisson point processes from the point of view of optimal transport and Ricci lower bounds. We construct a Riemannian structure on the space of point processes and the associated distance W that corresponds to the Benamou–Brenier variational formula. Our main tool is a non-local continuity equation formulated with the difference operator. The closure of the domain of the relative entropy is a complete geodesic space, when endowed with W. The geometry of this non-local infinite-dimensional space is analogous to that of spaces with positive Ricci curvature. Among others: (a) the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has an entropic Ricci curvature bounded from below by 1; (c) W satisfies an HWI inequality.}, author = {Dello Schiavo, Lorenzo and Herry, Ronan and Suzuki, Kohei}, issn = {2270-518X}, journal = {Journal de l'Ecole Polytechnique - Mathematiques}, pages = {957--1010}, publisher = {Ecole Polytechnique}, title = {{Wasserstein geometry and Ricci curvature bounds for Poisson spaces}}, doi = {10.5802/jep.270}, volume = {11}, year = {2024}, } @article{18490, abstract = {For large classes of even-dimensional Riemannian manifolds (Formula presented.), we construct and analyze conformally invariant random fields. These centered Gaussian fields (Formula presented.), called co-polyharmonic Gaussian fields, are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: (Formula presented.). They share a fundamental quasi-invariance property under conformal transformations. In terms of the co-polyharmonic Gaussian field (Formula presented.), we define the Liouville Quantum Gravity measure, a random measure on (Formula presented.), heuristically given as (Formula presented.) and rigorously obtained as almost sure weak limit of the right-hand side with (Formula presented.) replaced by suitable regular approximations (Formula presented.). In terms on the Liouville Quantum Gravity measure, we define the Liouville Brownian motion on (Formula presented.) and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimension with an ansatz based on Branson's (Formula presented.) -curvature: we give a rigorous meaning to the Polyakov–Liouville measure (Formula presented.) and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2-dimensional Riemannian manifolds, all compact non-negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even-dimensional Riemannian manifold is admissible. Our results concerning the logarithmic divergence of the kernel (Formula presented.) rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary closed manifolds. }, author = {Dello Schiavo, Lorenzo and Herry, Ronan and Kopfer, Eva and Sturm, Karl Theodor}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {5}, publisher = {London Mathematical Society}, title = {{Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension}}, doi = {10.1112/jlms.70003}, volume = {110}, year = {2024}, } @article{13271, abstract = {In this paper, we prove the convexity of trace functionals (A,B,C)↦Tr|BpACq|s, for parameters (p, q, s) that are best possible, where B and C are any n-by-n positive-definite matrices, and A is any n-by-n matrix. We also obtain the monotonicity versions of trace functionals of this type. As applications, we extend some results in Carlen et al. (Linear Algebra Appl 490:174–185, 2016), Hiai and Petz (Publ Res Inst Math Sci 48(3):525-542, 2012) and resolve a conjecture in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) in the matrix setting. Other conjectures in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) will also be discussed. We also show that some related trace functionals are not concave in general. Such concavity results were expected to hold in different problems.}, author = {Zhang, Haonan}, issn = {1424-0637}, journal = {Annales Henri Poincare}, pages = {2087--2106}, publisher = {Springer Nature}, title = {{Some convexity and monotonicity results of trace functionals}}, doi = {10.1007/s00023-023-01345-7}, volume = {25}, year = {2024}, } @article{13318, abstract = {Bohnenblust–Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree (Defant et al. in Math Ann 374(1):653–680, 2019). Such inequalities have found great applications in learning low-degree Boolean functions (Eskenazis and Ivanisvili in Proceedings of the 54th annual ACM SIGACT symposium on theory of computing, pp 203–207, 2022). Motivated by learning quantum observables, a qubit analogue of Bohnenblust–Hille inequality for Boolean cubes was recently conjectured in Rouzé et al. (Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum Boolean functions, 2022. arXiv preprint arXiv:2209.07279). The conjecture was resolved in Huang et al. (Learning to predict arbitrary quantum processes, 2022. arXiv preprint arXiv:2210.14894). In this paper, we give a new proof of these Bohnenblust–Hille inequalities for qubit system with constants that are dimension-free and of exponential growth in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials. Using similar ideas, we also study learning problems of low degree quantum observables and Bohr’s radius phenomenon on quantum Boolean cubes.}, author = {Volberg, Alexander and Zhang, Haonan}, issn = {1432-1807}, journal = {Mathematische Annalen}, pages = {1657--1676}, publisher = {Springer Nature}, title = {{Noncommutative Bohnenblust–Hille inequalities}}, doi = {10.1007/s00208-023-02680-0}, volume = {389}, year = {2024}, } @inproceedings{18897, abstract = {Score-based generative models (SGMs) are powerful tools to sample from complex data distributions. Their underlying idea is to (i) run a forward process for time T1 by adding noise to the data, (ii) estimate its score function, and (iii) use such estimate to run a reverse process. As the reverse process is initialized with the stationary distribution of the forward one, the existing analysis paradigm requires T1→∞. This is however problematic: from a theoretical viewpoint, for a given precision of the score approximation, the convergence guarantee fails as T1 diverges; from a practical viewpoint, a large T1 increases computational costs and leads to error propagation. This paper addresses the issue by considering a version of the popular predictor-corrector scheme: after running the forward process, we first estimate the final distribution via an inexact Langevin dynamics and then revert the process. Our key technical contribution is to provide convergence guarantees which require to run the forward process only for a fixed finite time T1. Our bounds exhibit a mild logarithmic dependence on the input dimension and the subgaussian norm of the target distribution, have minimal assumptions on the data, and require only to control the L2 loss on the score approximation, which is the quantity minimized in practice.}, author = {Pedrotti, Francesco and Maas, Jan and Mondelli, Marco}, booktitle = {Transactions on Machine Learning Research}, issn = {2835-8856}, title = {{Improved convergence of score-based diffusion models via prediction-correction}}, year = {2024}, }