@unpublished{20570,
  abstract     = {We investigate the minimal error in approximating a general probability
measure $\mu$ on $\mathbb{R}^d$ by the uniform measure on a finite set with
prescribed cardinality $n$. The error is measured in the $p$-Wasserstein
distance. In particular, when $1\le p<d$, we establish asymptotic upper and
lower bounds as $n \to \infty$ on the rescaled minimal error that have the
same, explicit dependency on $\mu$.
  In some instances, we prove that the rescaled minimal error has a limit.
These include general measures in dimension $d = 2$ with $1 \le p < 2$, and
uniform measures in arbitrary dimension with $1 \le p < d$. For some uniform
measures, we prove the limit existence for $p \ge d$ as well.
  For a class of compactly supported measures with H\"older densities, we
determine the convergence speed of the minimal error for every $p \ge 1$.
  Furthermore, we establish a new Pierce-type (i.e., nonasymptotic) upper
estimate of the minimal error when $1 \le p < d$.
  In the initial sections, we survey the state of the art and draw connections
with similar problems, such as classical and random quantization.},
  author       = {Quattrocchi, Filippo},
  booktitle    = {arXiv},
  keywords     = {optimal empirical quantization, vector quantization, Wasserstein distance, semidiscrete optimal transport, Zador’s Theorem, Pierce’s Lemma},
  title        = {{Asymptotics for optimal empirical quantization of measures}},
  doi          = {10.48550/arXiv.2408.12924},
  year         = {2024},
}

@article{12087,
  abstract     = {Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet–Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz–Schur multipliers over group algebras and generalized depolarizing semigroups.},
  author       = {Wirth, Melchior and Zhang, Haonan},
  issn         = {1424-0637},
  journal      = {Annales Henri Poincare},
  pages        = {717--750},
  publisher    = {Springer Nature},
  title        = {{Curvature-dimension conditions for symmetric quantum Markov semigroups}},
  doi          = {10.1007/s00023-022-01220-x},
  volume       = {24},
  year         = {2023},
}

@article{12104,
  abstract     = {We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.},
  author       = {Dello Schiavo, Lorenzo and Wirth, Melchior},
  issn         = {1424-3202},
  journal      = {Journal of Evolution Equations},
  number       = {1},
  publisher    = {Springer Nature},
  title        = {{Ergodic decompositions of Dirichlet forms under order isomorphisms}},
  doi          = {10.1007/s00028-022-00859-7},
  volume       = {23},
  year         = {2023},
}

@article{12911,
  abstract     = {This paper establishes new connections between many-body quantum systems, One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the ground-state energy of a finite-dimensional composite quantum system at positive temperature as a non-commutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dual-primal solutions in such non-commutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the many-electron ground-state energy can be computed via a non-commutative version of Sinkhorn algorithm. Our approach allows to prove convergence and robustness of this algorithm, which, to our best knowledge, were unknown even in the two marginal case. Our methods are based on a priori estimates in the dual problem, which we believe to be of independent interest. Finally, the above results are extended in 1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on the problem.},
  author       = {Feliciangeli, Dario and Gerolin, Augusto and Portinale, Lorenzo},
  issn         = {1096-0783},
  journal      = {Journal of Functional Analysis},
  number       = {4},
  publisher    = {Elsevier},
  title        = {{A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature}},
  doi          = {10.1016/j.jfa.2023.109963},
  volume       = {285},
  year         = {2023},
}

@article{12959,
  abstract     = {This paper deals with the large-scale behaviour of dynamical optimal transport on Zd
-periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density. Our homogenisation result is derived from a Γ
-convergence result for action functionals on curves of measures, which we prove under very mild growth conditions on the energy density. We investigate the cell formula in several cases of interest, including finite-volume discretisations of the Wasserstein distance, where non-trivial limiting behaviour occurs.},
  author       = {Gladbach, Peter and Kopfer, Eva and Maas, Jan and Portinale, Lorenzo},
  issn         = {1432-0835},
  journal      = {Calculus of Variations and Partial Differential Equations},
  number       = {5},
  publisher    = {Springer Nature},
  title        = {{Homogenisation of dynamical optimal transport on periodic graphs}},
  doi          = {10.1007/s00526-023-02472-z},
  volume       = {62},
  year         = {2023},
}

@article{13145,
  abstract     = {We prove a characterization of the Dirichlet–Ferguson measure over an arbitrary finite diffuse measure space. We provide an interpretation of this characterization in analogy with the Mecke identity for Poisson point processes.},
  author       = {Dello Schiavo, Lorenzo and Lytvynov, Eugene},
  issn         = {1083-589X},
  journal      = {Electronic Communications in Probability},
  pages        = {1--12},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{A Mecke-type characterization of the Dirichlet–Ferguson measure}},
  doi          = {10.1214/23-ECP528},
  volume       = {28},
  year         = {2023},
}

@article{13177,
  abstract     = {In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds cannot be established.},
  author       = {Hua, Bobo and Keller, Matthias and Schwarz, Michael and Wirth, Melchior},
  issn         = {1088-6826},
  journal      = {Proceedings of the American Mathematical Society},
  number       = {8},
  pages        = {3401--3414},
  publisher    = {American Mathematical Society},
  title        = {{Sobolev-type inequalities and eigenvalue growth on graphs with finite measure}},
  doi          = {10.1090/proc/14361},
  volume       = {151},
  year         = {2023},
}

@article{13319,
  abstract     = {We prove that the generator of the L2 implementation of a KMS-symmetric quantum Markov semigroup can be expressed as the square of a derivation with values in a Hilbert bimodule, extending earlier results by Cipriani and Sauvageot for tracially symmetric semigroups and the second-named author for GNS-symmetric semigroups. This result hinges on the introduction of a new completely positive map on the algebra of bounded operators on the GNS Hilbert space. This transformation maps symmetric Markov operators to symmetric Markov operators and is essential to obtain the required inner product on the Hilbert bimodule.},
  author       = {Vernooij, Matthijs and Wirth, Melchior},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  pages        = {381--416},
  publisher    = {Springer Nature},
  title        = {{Derivations and KMS-symmetric quantum Markov semigroups}},
  doi          = {10.1007/s00220-023-04795-6},
  volume       = {403},
  year         = {2023},
}

@unpublished{14732,
  abstract     = {Fragmented landscapes pose a significant threat to the persistence of species as they are highly susceptible to heightened risk of extinction due to the combined effects of genetic and demographic factors such as genetic drift and demographic stochasticity. This paper explores the intricate interplay between genetic load and extinction risk within metapopulations with a focus on understanding the impact of eco-evolutionary feedback mechanisms. We distinguish between two models of selection: soft selection, characterised by subpopulations maintaining carrying capacity despite load, and hard selection, where load can significantly affect population size. Within the soft selection framework, we investigate the impact of gene flow on genetic load at a single locus, while also considering the effect of selection strength and dominance coefficient. We subsequently build on this to examine how gene flow influences both population size and load under hard selection as well as identify critical thresholds for metapopulation persistence. Our analysis employs the diffusion, semi-deterministic and effective migration approximations. Our findings reveal that under soft selection, even modest levels of migration can significantly alleviate the burden of load. In sharp contrast, with hard selection, a much higher degree of gene flow is required to mitigate load and prevent the collapse of the metapopulation. Overall, this study sheds light into the crucial role migration plays in shaping the dynamics of genetic load and extinction risk in fragmented landscapes, offering valuable insights for conservation strategies and the preservation of diversity in a changing world.},
  author       = {Olusanya, Oluwafunmilola O and Khudiakova, Kseniia and Sachdeva, Himani},
  booktitle    = {bioRxiv},
  title        = {{Genetic load, eco-evolutionary feedback and extinction in a metapopulation}},
  doi          = {10.1101/2023.12.02.569702},
  year         = {2023},
}

@unpublished{20572,
  abstract     = {We present an elementary non-recursive formula for the multivariate moments
of the Dirichlet distribution on the standard simplex, in terms of the pattern
inventory of the moments' exponents. We obtain analog formulas for the
multivariate moments of the Dirichlet-Ferguson and Gamma measures. We further
introduce a polychromatic analogue of Ewens sampling formula on colored integer
partitions, discuss its relation with suitable extensions of Hoppe's urn model
and of the Chinese restaurant process, and prove that it satisfies an adapted
notion of consistency in the sense of Kingman.},
  author       = {Dello Schiavo, Lorenzo and Quattrocchi, Filippo},
  booktitle    = {arXiv},
  keywords     = {Dirichlet distribution, Ewens sampling formula, Hoppe urn model, colored partitions},
  title        = {{Multivariate Dirichlet moments and a polychromatic Ewens sampling formula}},
  doi          = {10.48550/arXiv.2309.11292},
  year         = {2023},
}

@article{10145,
  abstract     = {We study direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.},
  author       = {Dello Schiavo, Lorenzo},
  issn         = {1572-929X},
  journal      = {Potential Analysis},
  pages        = {573--615},
  publisher    = {Springer Nature},
  title        = {{Ergodic decomposition of Dirichlet forms via direct integrals and applications}},
  doi          = {10.1007/s11118-021-09951-y},
  volume       = {58},
  year         = {2023},
}

@unpublished{17351,
  abstract     = {Contractive coupling rates have been recently introduced by Conforti as a
tool to establish convex Sobolev inequalities (including modified log-Sobolev
and Poincar\'{e} inequality) for some classes of Markov chains. In this work,
we show how contractive coupling rates can also be used to prove stronger
inequalities, in the form of curvature lower bounds for Markov chains and
geodesic convexity of entropic functionals. We illustrate this in several
examples discussed by Conforti, where in particular, after appropriately
choosing a parameter function, we establish positive curvature in the entropic
and (discrete) Bakry--\'{E}mery sense. 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},
  booktitle    = {arXiv},
  title        = {{Contractive coupling rates and curvature lower bounds for Markov chains}},
  doi          = {10.48550/arXiv.2308.00516},
  year         = {2023},
}

@article{11330,
  abstract     = {In this article we study the noncommutative transport distance introduced by Carlen and Maas and its entropic regularization defined by Becker and Li. We prove a duality formula that can be understood as a quantum version of the dual Benamou–Brenier formulation of the Wasserstein distance in terms of subsolutions of a Hamilton–Jacobi–Bellmann equation.},
  author       = {Wirth, Melchior},
  issn         = {1572-9613},
  journal      = {Journal of Statistical Physics},
  number       = {2},
  publisher    = {Springer Nature},
  title        = {{A dual formula for the noncommutative transport distance}},
  doi          = {10.1007/s10955-022-02911-9},
  volume       = {187},
  year         = {2022},
}

@article{11354,
  abstract     = {We construct a recurrent diffusion process with values in the space of probability measures over an arbitrary closed Riemannian manifold of dimension d≥2. The process is associated with the Dirichlet form defined by integration of the Wasserstein gradient w.r.t. the Dirichlet–Ferguson measure, and is the counterpart on multidimensional base spaces to the modified massive Arratia flow over the unit interval described in V. Konarovskyi and M.-K. von Renesse (Comm. Pure Appl. Math. 72 (2019) 764–800). Together with two different constructions of the process, we discuss its ergodicity, invariant sets, finite-dimensional approximations, and Varadhan short-time asymptotics.},
  author       = {Dello Schiavo, Lorenzo},
  issn         = {2168-894X},
  journal      = {Annals of Probability},
  number       = {2},
  pages        = {591--648},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold}},
  doi          = {10.1214/21-AOP1541},
  volume       = {50},
  year         = {2022},
}

@article{11447,
  abstract     = {Empirical essays of fitness landscapes suggest that they may be rugged, that is having multiple fitness peaks. Such fitness landscapes, those that have multiple peaks, necessarily have special local structures, called reciprocal sign epistasis (Poelwijk et al. in J Theor Biol 272:141–144, 2011). Here, we investigate the quantitative relationship between the number of fitness peaks and the number of reciprocal sign epistatic interactions. Previously, it has been shown (Poelwijk et al. in J Theor Biol 272:141–144, 2011) that pairwise reciprocal sign epistasis is a necessary but not sufficient condition for the existence of multiple peaks. Applying discrete Morse theory, which to our knowledge has never been used in this context, we extend this result by giving the minimal number of reciprocal sign epistatic interactions required to create a given number of peaks.},
  author       = {Saona Urmeneta, Raimundo J and Kondrashov, Fyodor and Khudiakova, Kseniia},
  issn         = {1522-9602},
  journal      = {Bulletin of Mathematical Biology},
  keywords     = {Computational Theory and Mathematics, General Agricultural and Biological Sciences, Pharmacology, General Environmental Science, General Biochemistry, Genetics and Molecular Biology, General Mathematics, Immunology, General Neuroscience},
  number       = {8},
  publisher    = {Springer Nature},
  title        = {{Relation between the number of peaks and the number of reciprocal sign epistatic interactions}},
  doi          = {10.1007/s11538-022-01029-z},
  volume       = {84},
  year         = {2022},
}

@article{11700,
  abstract     = {This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou–Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan–Kinderlehrer–Otto, we show that McKean–Vlasov equations can be formulated as gradient flow of the free energy in the Wasserstein space of probability measures. The proofs of these results are based on careful regularisation arguments to circumvent some of the difficulties arising in metric graphs, namely, branching of geodesics and the failure of semi-convexity of entropy functionals in the Wasserstein space.},
  author       = {Erbar, Matthias and Forkert, Dominik L and Maas, Jan and Mugnolo, Delio},
  issn         = {1556-181X},
  journal      = {Networks and Heterogeneous Media},
  number       = {5},
  pages        = {687--717},
  publisher    = {American Institute of Mathematical Sciences},
  title        = {{Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph}},
  doi          = {10.3934/nhm.2022023},
  volume       = {17},
  year         = {2022},
}

@article{11739,
  abstract     = {We consider finite-volume approximations of Fokker--Planck equations on bounded convex domains in $\mathbb{R}^d$ and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker--Planck equation via the method of evolutionary $\Gamma$-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalizing the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.},
  author       = {Forkert, Dominik L and Maas, Jan and Portinale, Lorenzo},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {Fokker--Planck equation, gradient flow, evolutionary $\Gamma$-convergence},
  number       = {4},
  pages        = {4297--4333},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions}},
  doi          = {10.1137/21M1410968},
  volume       = {54},
  year         = {2022},
}

@article{11916,
  abstract     = {A domain is called Kac regular for a quadratic form on L2 if every functions vanishing almost everywhere outside the domain can be approximated in form norm by functions with compact support in the domain. It is shown that this notion is stable under domination of quadratic forms. As applications measure perturbations of quasi-regular Dirichlet forms, Cheeger energies on metric measure spaces and Schrödinger operators on manifolds are studied. Along the way a characterization of the Sobolev space with Dirichlet boundary conditions on domains in infinitesimally Riemannian metric measure spaces is obtained.},
  author       = {Wirth, Melchior},
  issn         = {2538-225X},
  journal      = {Advances in Operator Theory},
  keywords     = {Algebra and Number Theory, Analysis},
  number       = {3},
  publisher    = {Springer Nature},
  title        = {{Kac regularity and domination of quadratic forms}},
  doi          = {10.1007/s43036-022-00199-w},
  volume       = {7},
  year         = {2022},
}

@article{12177,
  abstract     = {Using elementary hyperbolic geometry, we give an explicit formula for the contraction constant of the skinning map over moduli spaces of relatively acylindrical hyperbolic manifolds.},
  author       = {Cremaschi, Tommaso and Dello Schiavo, Lorenzo},
  issn         = {2330-1511},
  journal      = {Proceedings of the American Mathematical Society, Series B},
  number       = {43},
  pages        = {445--459},
  publisher    = {American Mathematical Society},
  title        = {{Effective contraction of Skinning maps}},
  doi          = {10.1090/bproc/134},
  volume       = {9},
  year         = {2022},
}

@article{12210,
  abstract     = {The aim of this paper is to find new estimates for the norms of functions of a (minus) distinguished Laplace operator L on the ‘ax+b’ groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type ψ(L−−√)exp(itL−−√), with ψ∈C0(R). We show that for t→+∞, the convolution kernel kt of this operator satisfies
∥kt∥1≍t,∥kt∥∞≍1,
so that the upper estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary component, we recall the Plancherel density of L and spend certain time presenting and comparing different approaches to its calculation. Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami operator Δ~, closely related to L. The functions include in particular exp(−tΔ~γ), t>0,γ>0, and (Δ~−z)s, with complex z, s.},
  author       = {Akylzhanov, Rauan and Kuznetsova, Yulia and Ruzhansky, Michael and Zhang, Haonan},
  issn         = {1432-1823},
  journal      = {Mathematische Zeitschrift},
  keywords     = {General Mathematics},
  number       = {4},
  pages        = {2327--2352},
  publisher    = {Springer Nature},
  title        = {{Norms of certain functions of a distinguished Laplacian on the ax + b groups}},
  doi          = {10.1007/s00209-022-03143-z},
  volume       = {302},
  year         = {2022},
}