@article{12216, abstract = {Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.}, author = {Carlen, Eric A. and Zhang, Haonan}, issn = {0024-3795}, journal = {Linear Algebra and its Applications}, keywords = {Discrete Mathematics and Combinatorics, Geometry and Topology, Numerical Analysis, Algebra and Number Theory}, pages = {289--310}, publisher = {Elsevier}, title = {{Monotonicity versions of Epstein's concavity theorem and related inequalities}}, doi = {10.1016/j.laa.2022.09.001}, volume = {654}, year = {2022}, } @article{12281, abstract = {We study the hydrodynamic and hydrostatic limits of the one-dimensional open symmetric inclusion process with slow boundary. Depending on the value of the parameter tuning the interaction rate of the bulk of the system with the boundary, we obtain a linear heat equation with either Dirichlet, Robin or Neumann boundary conditions as hydrodynamic equation. In our approach, we combine duality and first-second class particle techniques to reduce the scaling limit of the inclusion process to the limiting behavior of a single, non-interacting, particle.}, author = {Franceschini, Chiara and Gonçalves, Patrícia and Sau, Federico}, issn = {1350-7265}, journal = {Bernoulli}, keywords = {Statistics and Probability}, number = {2}, pages = {1340--1381}, publisher = {Bernoulli Society for Mathematical Statistics and Probability}, title = {{Symmetric inclusion process with slow boundary: Hydrodynamics and hydrostatics}}, doi = {10.3150/21-bej1390}, volume = {28}, year = {2022}, } @article{10797, abstract = {We consider symmetric partial exclusion and inclusion processes in a general graph in contact with reservoirs, where we allow both for edge disorder and well-chosen site disorder. We extend the classical dualities to this context and then we derive new orthogonal polynomial dualities. From the classical dualities, we derive the uniqueness of the non-equilibrium steady state and obtain correlation inequalities. Starting from the orthogonal polynomial dualities, we show universal properties of n-point correlation functions in the non-equilibrium steady state for systems with at most two different reservoir parameters, such as a chain with reservoirs at left and right ends.}, author = {Floreani, Simone and Redig, Frank and Sau, Federico}, issn = {0246-0203}, journal = {Annales de l'institut Henri Poincare (B) Probability and Statistics}, number = {1}, pages = {220--247}, publisher = {Institute of Mathematical Statistics}, title = {{Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations}}, doi = {10.1214/21-AIHP1163}, volume = {58}, 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{10023, abstract = {We study the temporal dissipation of variance and relative entropy for ergodic Markov Chains in continuous time, and compute explicitly the corresponding dissipation rates. These are identified, as is well known, in the case of the variance in terms of an appropriate Hilbertian norm; and in the case of the relative entropy, in terms of a Dirichlet form which morphs into a version of the familiar Fisher information under conditions of detailed balance. Here we obtain trajectorial versions of these results, valid along almost every path of the random motion and most transparent in the backwards direction of time. Martingale arguments and time reversal play crucial roles, as in the recent work of Karatzas, Schachermayer and Tschiderer for conservative diffusions. Extensions are developed to general “convex divergences” and to countable state-spaces. The steepest descent and gradient flow properties for the variance, the relative entropy, and appropriate generalizations, are studied along with their respective geometries under conditions of detailed balance, leading to a very direct proof for the HWI inequality of Otto and Villani in the present context.}, author = {Karatzas, Ioannis and Maas, Jan and Schachermayer, Walter}, issn = {1526-7555}, journal = {Communications in Information and Systems}, keywords = {Markov Chain, relative entropy, time reversal, steepest descent, gradient flow}, number = {4}, pages = {481--536}, publisher = {International Press}, title = {{Trajectorial dissipation and gradient flow for the relative entropy in Markov chains}}, doi = {10.4310/CIS.2021.v21.n4.a1}, volume = {21}, year = {2021}, } @article{10613, abstract = {Motivated by the recent preprint [\emph{arXiv:2004.08412}] by Ayala, Carinci, and Redig, we first provide a general framework for the study of scaling limits of higher-order fields. Then, by considering the same class of infinite interacting particle systems as in [\emph{arXiv:2004.08412}], namely symmetric simple exclusion and inclusion processes in the d-dimensional Euclidean lattice, we prove the hydrodynamic limit, and convergence for the equilibrium fluctuations, of higher-order fields. In particular, the limit fields exhibit a tensor structure. Our fluctuation result differs from that in [\emph{arXiv:2004.08412}], since we considered-dimensional Euclidean lattice, we prove the hydrodynamic limit, and convergence for the equilibrium fluctuations, of higher-order fields. In particular, the limit fields exhibit a tensor structure. Our fluctuation result differs from that in [\emph{arXiv:2004.08412}], since we consider a different notion of higher-order fluctuation fields.}, author = {Chen, Joe P. and Sau, Federico}, issn = {1024-2953}, journal = {Markov Processes And Related Fields}, keywords = {interacting particle systems, higher-order fields, hydrodynamic limit, equilibrium fluctuations, duality}, number = {3}, pages = {339--380}, publisher = {Polymat Publishing}, title = {{Higher-order hydrodynamics and equilibrium fluctuations of interacting particle systems}}, volume = {27}, year = {2021}, } @article{9973, abstract = {In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.}, author = {Wirth, Melchior and Zhang, Haonan}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, pages = {761–791}, publisher = {Springer Nature}, title = {{Complete gradient estimates of quantum Markov semigroups}}, doi = {10.1007/s00220-021-04199-4}, volume = {387}, year = {2021}, } @article{10024, abstract = {In this paper, we introduce a random environment for the exclusion process in obtained by assigning a maximal occupancy to each site. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we derive a quenched hydrodynamic limit in path space by strengthening the mild solution approach initiated in Nagy (2002) and Faggionato (2007). To this purpose, we prove, employing the technology developed for the random conductance model, a homogenization result in the form of an arbitrary starting point quenched invariance principle for a single particle in the same environment, which is a result of independent interest. The self-duality property of the partial exclusion process allows us to transfer this homogenization result to the particle system and, then, apply the tightness criterion in Redig et al. (2020).}, author = {Floreani, Simone and Redig, Frank and Sau, Federico}, issn = {0304-4149}, journal = {Stochastic Processes and their Applications}, keywords = {hydrodynamic limit, random environment, random conductance model, arbitrary starting point quenched invariance principle, duality, mild solution}, pages = {124--158}, publisher = {Elsevier}, title = {{Hydrodynamics for the partial exclusion process in random environment}}, doi = {10.1016/j.spa.2021.08.006}, volume = {142}, year = {2021}, } @article{10070, abstract = {We extensively discuss the Rademacher and Sobolev-to-Lipschitz properties for generalized intrinsic distances on strongly local Dirichlet spaces possibly without square field operator. We present many non-smooth and infinite-dimensional examples. As an application, we prove the integral Varadhan short-time asymptotic with respect to a given distance function for a large class of strongly local Dirichlet forms.}, author = {Dello Schiavo, Lorenzo and Suzuki, Kohei}, issn = {1096-0783}, journal = {Journal of Functional Analysis}, number = {11}, publisher = {Elsevier}, title = {{Rademacher-type theorems and Sobolev-to-Lipschitz properties for strongly local Dirichlet spaces}}, doi = {10.1016/j.jfa.2021.109234}, volume = {281}, year = {2021}, } @article{9627, abstract = {We compute the deficiency spaces of operators of the form 𝐻𝐴⊗̂ 𝐼+𝐼⊗̂ 𝐻𝐵, for symmetric 𝐻𝐴 and self-adjoint 𝐻𝐵. This enables us to construct self-adjoint extensions (if they exist) by means of von Neumann's theory. The structure of the deficiency spaces for this case was asserted already in Ibort et al. [Boundary dynamics driven entanglement, J. Phys. A: Math. Theor. 47(38) (2014) 385301], but only proven under the restriction of 𝐻𝐵 having discrete, non-degenerate spectrum.}, author = {Lenz, Daniel and Weinmann, Timon and Wirth, Melchior}, issn = {1464-3839}, journal = {Proceedings of the Edinburgh Mathematical Society}, number = {3}, pages = {443--447}, publisher = {Cambridge University Press}, title = {{Self-adjoint extensions of bipartite Hamiltonians}}, doi = {10.1017/S0013091521000080}, volume = {64}, year = {2021}, } @phdthesis{10030, abstract = {This PhD thesis is primarily focused on the study of discrete transport problems, introduced for the first time in the seminal works of Maas [Maa11] and Mielke [Mie11] on finite state Markov chains and reaction-diffusion equations, respectively. More in detail, my research focuses on the study of transport costs on graphs, in particular the convergence and the stability of such problems in the discrete-to-continuum limit. This thesis also includes some results concerning non-commutative optimal transport. The first chapter of this thesis consists of a general introduction to the optimal transport problems, both in the discrete, the continuous, and the non-commutative setting. Chapters 2 and 3 present the content of two works, obtained in collaboration with Peter Gladbach, Eva Kopfer, and Jan Maas, where we have been able to show the convergence of discrete transport costs on periodic graphs to suitable continuous ones, which can be described by means of a homogenisation result. We first focus on the particular case of quadratic costs on the real line and then extending the result to more general costs in arbitrary dimension. Our results are the first complete characterisation of limits of transport costs on periodic graphs in arbitrary dimension which do not rely on any additional symmetry. In Chapter 4 we turn our attention to one of the intriguing connection between evolution equations and optimal transport, represented by the theory of gradient flows. We show that discrete gradient flow structures associated to a finite volume approximation of a certain class of diffusive equations (Fokker–Planck) is stable in the limit of vanishing meshes, reproving the convergence of the scheme via the method of evolutionary Γ-convergence and exploiting a more variational point of view on the problem. This is based on a collaboration with Dominik Forkert and Jan Maas. Chapter 5 represents a change of perspective, moving away from the discrete world and reaching the non-commutative one. As in the discrete case, we discuss how classical tools coming from the commutative optimal transport can be translated into the setting of density matrices. In particular, in this final chapter we present a non-commutative version of the Schrödinger problem (or entropic regularised optimal transport problem) and discuss existence and characterisation of minimisers, a duality result, and present a non-commutative version of the well-known Sinkhorn algorithm to compute the above mentioned optimisers. This is based on a joint work with Dario Feliciangeli and Augusto Gerolin. Finally, Appendix A and B contain some additional material and discussions, with particular attention to Harnack inequalities and the regularity of flows on discrete spaces.}, author = {Portinale, Lorenzo}, issn = {2663-337X}, publisher = {Institute of Science and Technology Austria}, title = {{Discrete-to-continuum limits of transport problems and gradient flows in the space of measures}}, doi = {10.15479/at:ista:10030}, year = {2021}, } @unpublished{9792, 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 careful 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}, booktitle = {arXiv}, title = {{A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature}}, doi = {10.48550/arXiv.2106.11217}, year = {2021}, } @phdthesis{9733, abstract = {This thesis is the result of the research carried out by the author during his PhD at IST Austria between 2017 and 2021. It mainly focuses on the Fröhlich polaron model, specifically to its regime of strong coupling. This model, which is rigorously introduced and discussed in the introduction, has been of great interest in condensed matter physics and field theory for more than eighty years. It is used to describe an electron interacting with the atoms of a solid material (the strength of this interaction is modeled by the presence of a coupling constant α in the Hamiltonian of the system). The particular regime examined here, which is mathematically described by considering the limit α →∞, displays many interesting features related to the emergence of classical behavior, which allows for a simplified effective description of the system under analysis. The properties, the range of validity and a quantitative analysis of the precision of such classical approximations are the main object of the present work. We specify our investigation to the study of the ground state energy of the system, its dynamics and its effective mass. For each of these problems, we provide in the introduction an overview of the previously known results and a detailed account of the original contributions by the author.}, author = {Feliciangeli, Dario}, issn = {2663-337X}, pages = {180}, publisher = {Institute of Science and Technology Austria}, title = {{The polaron at strong coupling}}, doi = {10.15479/at:ista:9733}, year = {2021}, } @article{6358, abstract = {We study dynamical optimal transport metrics between density matricesassociated to symmetric Dirichlet forms on finite-dimensional C∗-algebras. Our settingcovers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, andspectral gap estimates.}, author = {Carlen, Eric A. and Maas, Jan}, issn = {15729613}, journal = {Journal of Statistical Physics}, number = {2}, pages = {319--378}, publisher = {Springer Nature}, title = {{Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems}}, doi = {10.1007/s10955-019-02434-w}, volume = {178}, year = {2020}, } @inbook{74, abstract = {We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian measures.}, author = {Akopyan, Arseniy and Karasev, Roman}, booktitle = {Geometric Aspects of Functional Analysis}, editor = {Klartag, Bo'az and Milman, Emanuel}, isbn = {9783030360191}, issn = {16179692}, pages = {1--27}, publisher = {Springer Nature}, title = {{Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures}}, doi = {10.1007/978-3-030-36020-7_1}, volume = {2256}, year = {2020}, } @article{7388, abstract = {We give a Wong-Zakai type characterisation of the solutions of quasilinear heat equations driven by space-time white noise in 1 + 1 dimensions. In order to show that the renormalisation counterterms are local in the solution, a careful arrangement of a few hundred terms is required. The main tool in this computation is a general ‘integration by parts’ formula that provides a number of linear identities for the renormalisation constants.}, author = {Gerencser, Mate}, issn = {0294-1449}, journal = {Annales de l'Institut Henri Poincaré C, Analyse non linéaire}, number = {3}, pages = {663--682}, publisher = {Elsevier}, title = {{Nondivergence form quasilinear heat equations driven by space-time white noise}}, doi = {10.1016/j.anihpc.2020.01.003}, volume = {37}, year = {2020}, } @article{7509, abstract = {In this paper we study the joint convexity/concavity of the trace functions Ψp,q,s(A,B)=Tr(Bq2K∗ApKBq2)s, p,q,s∈R, where A and B are positive definite matrices and K is any fixed invertible matrix. We will give full range of (p,q,s)∈R3 for Ψp,q,s to be jointly convex/concave for all K. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of (α,z) for α-z Rényi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψp,0,1/p for 0= 1. Along the argument, we prove Mosco- and I-convergence results for discrete energy functionals, which are of independent interest for convergence of equivalent gradient flow structures in Hilbert spaces. The second part investigates L2-Wasserstein flows on metric graph. The starting point is a Benamou-Brenier formula for the L2-Wasserstein distance, which is proved via a regularisation scheme for solutions of the continuity equation, adapted to the peculiar geometric structure of metric graphs. Based on those results, we show that the L2-Wasserstein space over a metric graph admits a gradient flow which may be identified as a solution of a Fokker-Planck equation. In the third part, we focus again on the discrete gradient flows, already encountered in the first part. We propose a variational structure which extends the gradient flow structure to Markov chains violating the detailed-balance conditions. Using this structure, we characterise contraction estimates for the discrete heat flow in terms of convexity of corresponding path-dependent energy functionals. In addition, we use this approach to derive several functional inequalities for said functionals.}, author = {Forkert, Dominik L}, issn = {2663-337X}, pages = {154}, publisher = {Institute of Science and Technology Austria}, title = {{Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains}}, doi = {10.15479/AT:ISTA:7629}, year = {2020}, }