@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{21894, abstract = {The Dean–Kawasaki equation—one of the most fundamental SPDEs of fluctuating hydrodynamics—has been proposed as a model for density fluctuations in weakly interacting particle systems. In its original form, it is highly singular and fails to be renormalizable, even by approaches such as regularity structures and paracontrolled distributions, hindering mathematical approaches to its rigorous justification. It has been understood recently that it is natural to introduce a suitable regularization, for example, by applying a formal spatial discretization or by truncating high-frequency noise: This yields well-posed equations that should still precisely approximate the law of the particle density fluctuations. In the present work, we prove that a regularization in the form of a formal discretization of the Dean–Kawasaki equation indeed accurately describes density fluctuations in systems of weakly interacting diffusing particles: We show that, in suitable weak metrics, the law of fluctuations as predicted by the discretized Dean–Kawasaki SPDE approximates the law of fluctuations of the original particle system, up to an error that is of arbitrarily high order in the inverse particle number and a discretization error. In particular, the Dean– Kawasaki equation provides a means for efficient and accurate simulations of density fluctuations in weakly interacting particle systems.}, author = {Cornalba, Federico and Fischer, Julian L and Ingmanns, Jonas and Raithel, Claudia}, issn = {2168-894X}, journal = {The Annals of Probability}, keywords = {Weakly interacting particle systems, fluctuating hydrodynamics, Dean-Kawasaki equation, stochastic PDEs, numerical approximation}, number = {1}, pages = {155--215}, publisher = {Institute of Mathematical Statistics}, title = {{Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation}}, doi = {10.1214/25-aop1763}, volume = {54}, year = {2026}, } @phdthesis{21918, author = {Khudiakova, Kseniia}, issn = {2663-337X}, pages = {89}, publisher = {Institute of Science and Technology Austria}, title = {{How epistasis and purifying selection shape genetic diversity}}, doi = {10.15479/AT-ISTA-21918}, year = {2026}, } @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}, keywords = {Dirichlet forms, Domination of semigroups, Dirichlet, Neumann and Robin boundary conditions}, number = {1}, 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 = {2026}, } @article{18706, abstract = {We prove discrete-to-continuum convergence for dynamical optimal transport on Zd -periodic graphs with cost functional having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of 1 -Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations.}, author = {Portinale, Lorenzo and Quattrocchi, Filippo}, issn = {1469-4425}, journal = {European Journal of Applied Mathematics}, keywords = {optimal transport, discrete-to-continuum, homogenisation, linear growth, gamma-convergence}, number = {3}, pages = {614--642}, publisher = {Cambridge University Press}, title = {{Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs}}, doi = {10.1017/s0956792524000810}, volume = {37}, 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{19027, abstract = {Stochastic PDEs of fluctuating hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a multilevel Monte Carlo (MLMC) scheme for the Dean–Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically and demonstrate numerically that our MLMC scheme provides a significant reduction in computational cost (with respect to a standard Monte Carlo method) in the simulation of the Dean–Kawasaki equation. Specifically, we link this reduction in cost to having a sufficiently large average particle density and show that sizeable cost reductions can be obtained even when we have solutions with regions of low density. Numerical simulations are provided in the two-dimensional case, confirming our theoretical predictions. Our results are formulated entirely in terms of the law of distributions rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups altogether despite the Dean–Kawasaki equation being highly singular.}, author = {Cornalba, Federico and Fischer, Julian L}, issn = {1095-7170}, journal = {SIAM Journal on Numerical Analysis}, number = {1}, pages = {262--287}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics}}, doi = {10.1137/23M1617345}, volume = {63}, 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{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}, } @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{14451, abstract = {We investigate the potential of Multi-Objective, Deep Reinforcement Learning for stock and cryptocurrency single-asset trading: in particular, we consider a Multi-Objective algorithm which generalizes the reward functions and discount factor (i.e., these components are not specified a priori, but incorporated in the learning process). Firstly, using several important assets (BTCUSD, ETHUSDT, XRPUSDT, AAPL, SPY, NIFTY50), we verify the reward generalization property of the proposed Multi-Objective algorithm, and provide preliminary statistical evidence showing increased predictive stability over the corresponding Single-Objective strategy. Secondly, we show that the Multi-Objective algorithm has a clear edge over the corresponding Single-Objective strategy when the reward mechanism is sparse (i.e., when non-null feedback is infrequent over time). Finally, we discuss the generalization properties with respect to the discount factor. The entirety of our code is provided in open-source format.}, author = {Cornalba, Federico and Disselkamp, Constantin and Scassola, Davide and Helf, Christopher}, issn = {1433-3058}, journal = {Neural Computing and Applications}, number = {2}, pages = {617--637}, publisher = {Springer Nature}, title = {{Multi-objective reward generalization: Improving performance of Deep Reinforcement Learning for applications in single-asset trading}}, doi = {10.1007/s00521-023-09033-7}, volume = {36}, year = {2024}, } @article{14884, abstract = {We perform a stochastic homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional defined on magnetizations taking value in the unit sphere and including both symmetric and antisymmetric exchange contributions. This Gamma-limit corresponds to a micromagnetic energy functional with homogeneous coefficients. We provide explicit formulas for the effective magnetic properties of the composite material in terms of homogenization correctors. Additionally, the variational analysis of the two exchange energy terms is performed in the more general setting of functionals defined on manifold-valued maps with Sobolev regularity, in the case in which the target manifold is a bounded, orientable smooth surface with tubular neighborhood of uniform thickness. Eventually, we present an explicit characterization of minimizers of the effective exchange in the case of magnetic multilayers, providing quantitative evidence of Dzyaloshinskii’s predictions on the emergence of helical structures in composite ferromagnetic materials with stochastic microstructure.}, author = {Davoli, Elisa and D’Elia, Lorenza and Ingmanns, Jonas}, issn = {1432-1467}, journal = {Journal of Nonlinear Science}, number = {2}, publisher = {Springer Nature}, title = {{Stochastic homogenization of micromagnetic energies and emergence of magnetic skyrmions}}, doi = {10.1007/s00332-023-10005-3}, volume = {34}, year = {2024}, } @article{14934, abstract = {We study random perturbations of a Riemannian manifold (M, g) by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields h• : ω → hω will act on the manifold via the conformal transformation g → gω := e2hω g. Our focus will be on the regular case with Hurst parameter H > 0, the critical case H = 0 being the celebrated Liouville geometry in two dimensions. We want to understand how basic geometric and functional-analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise? Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.}, author = {Dello Schiavo, Lorenzo and Kopfer, Eva and Sturm, Karl Theodor}, issn = {1572-929X}, journal = {Potential Analysis}, pages = {501--553}, publisher = {Springer Nature}, title = {{A discovery tour in random Riemannian geometry}}, doi = {10.1007/s11118-023-10118-0}, volume = {61}, year = {2024}, } @article{15317, abstract = {We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain Ω, with both fast and slow boundary. For the random walks on Ω dual to SEP/SIP we establish: a functional-CLT-type convergence to the Brownian motion on Ω with either Neumann (slow boundary), Dirichlet (fast boundary), or Robin (at criticality) boundary conditions; the discrete-to-continuum convergence of the corresponding harmonic profiles. As a consequence, we rigorously derive the hydrodynamic and hydrostatic limits for SEP/SIP on Ω, and analyze their stationary nonequilibrium fluctuations. All scaling limit results for SEP/SIP concern finite-dimensional distribution convergence only, as our duality techniques do not require to establish tightness for the fields associated to the particle systems.}, author = {Dello Schiavo, Lorenzo and Portinale, Lorenzo and Sau, Federico}, issn = {1050-5164}, journal = {Annals of Applied Probability}, number = {2}, pages = {1789--1845}, publisher = {Institute of Mathematical Statistics}, title = {{Scaling limits of random walks, harmonic profiles, and stationary nonequilibrium states in Lipschitz domains}}, doi = {10.1214/23-AAP2007}, volume = {34}, year = {2024}, } @article{17143, abstract = {This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on Rn. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on Rn. .}, author = {Dello Schiavo, Lorenzo and Maas, Jan and Pedrotti, Francesco}, issn = {1088-6850}, journal = {Transactions of the American Mathematical Society}, number = {6}, pages = {3779--3804}, publisher = {American Mathematical Society}, title = {{Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces}}, doi = {10.1090/tran/9156}, volume = {377}, 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}, } @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}, } @article{17282, abstract = {Let X be a vector field and Y be a co-vector field on a smooth manifold M. Does there exist a smooth Riemannian metric gαβ on M such that Yβ=gαβXα ? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we show that a finite-dimensional ergodic Lindblad equation admits a gradient flow structure for the von Neumann relative entropy if and only if the condition of BKM-detailed balance holds.}, author = {Brooks, Morris and Maas, Jan}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {6}, publisher = {Springer Nature}, title = {{Characterisation of gradient flows for a given functional}}, doi = {10.1007/s00526-024-02755-z}, volume = {63}, year = {2024}, } @phdthesis{17336, abstract = {This thesis deals with the study of stochastic processes and their ergodicity properties. The variety of problems encountered calls for a set of different approaches, ranging from classical to modern ones: a special place is held by probabilistic methods based on couplings, by functional inequalities, and by the theory of gradient flows in the space of measures. The material is organized as follows. Chapter 1 contains the introduction to this thesis, starting with a general presentation of some of the relevant topics. Section 1.1 is dedicated to the theory of gradient flows in metric spaces, and introduces the first contribution of this thesis [DSMP24], which is presented in detail in Chapter 2. Section 1.2 moves to the topic of curvature of Markov chains, concluding with a brief description of our second contribution [Ped23], which is included in Chapter 3. Section 1.3 discusses applications of stochastic processes to the theory of sampling, in particular the recent framework of score-based diffusion models, and our contribution [PMM24], which is contained in Chapter 4. Section 1.4 discusses some related problems, concerning the regularization properties of the heat flow. It serves as a motivation for the work [BP24], which we report in Chapter 5. Finally, Section 1.5 discusses the last contribution of this thesis, which can be found in Chapter 6. It deals with the convergence to equilibrium of a particular stochastic model from quantitative genetics: this is established via some functional inequalities, which we prove with probabilistic arguments based on couplings. }, author = {Pedrotti, Francesco}, issn = {2663-337X}, pages = {183}, publisher = {Institute of Science and Technology Austria}, title = {{Functional inequalities and convergence of stochastic processes}}, doi = {10.15479/at:ista:17336}, year = {2024}, }