TY - JOUR AB - 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. AU - Carlen, Eric A. AU - Zhang, Haonan ID - 12216 JF - Linear Algebra and its Applications KW - Discrete Mathematics and Combinatorics KW - Geometry and Topology KW - Numerical Analysis KW - Algebra and Number Theory SN - 0024-3795 TI - Monotonicity versions of Epstein's concavity theorem and related inequalities VL - 654 ER - TY - JOUR AB - 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. AU - Franceschini, Chiara AU - Gonçalves, Patrícia AU - Sau, Federico ID - 12281 IS - 2 JF - Bernoulli KW - Statistics and Probability SN - 1350-7265 TI - Symmetric inclusion process with slow boundary: Hydrodynamics and hydrostatics VL - 28 ER - TY - JOUR AB - 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. AU - Floreani, Simone AU - Redig, Frank AU - Sau, Federico ID - 10797 IS - 1 JF - Annales de l'institut Henri Poincare (B) Probability and Statistics SN - 0246-0203 TI - Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations VL - 58 ER - TY - JOUR AB - 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. AU - Dello Schiavo, Lorenzo ID - 11354 IS - 2 JF - Annals of Probability SN - 0091-1798 TI - The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold VL - 50 ER - TY - JOUR AB - 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. AU - Karatzas, Ioannis AU - Maas, Jan AU - Schachermayer, Walter ID - 10023 IS - 4 JF - Communications in Information and Systems KW - Markov Chain KW - relative entropy KW - time reversal KW - steepest descent KW - gradient flow SN - 1526-7555 TI - Trajectorial dissipation and gradient flow for the relative entropy in Markov chains VL - 21 ER - TY - JOUR AB - 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. AU - Chen, Joe P. AU - Sau, Federico ID - 10613 IS - 3 JF - Markov Processes And Related Fields KW - interacting particle systems KW - higher-order fields KW - hydrodynamic limit KW - equilibrium fluctuations KW - duality SN - 1024-2953 TI - Higher-order hydrodynamics and equilibrium fluctuations of interacting particle systems VL - 27 ER - TY - JOUR AB - 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. AU - Wirth, Melchior AU - Zhang, Haonan ID - 9973 JF - Communications in Mathematical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0010-3616 TI - Complete gradient estimates of quantum Markov semigroups VL - 387 ER - TY - JOUR AB - 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). AU - Floreani, Simone AU - Redig, Frank AU - Sau, Federico ID - 10024 JF - Stochastic Processes and their Applications KW - hydrodynamic limit KW - random environment KW - random conductance model KW - arbitrary starting point quenched invariance principle KW - duality KW - mild solution SN - 0304-4149 TI - Hydrodynamics for the partial exclusion process in random environment VL - 142 ER - TY - JOUR AB - 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. AU - Dello Schiavo, Lorenzo AU - Suzuki, Kohei ID - 10070 IS - 11 JF - Journal of Functional Analysis SN - 0022-1236 TI - Rademacher-type theorems and Sobolev-to-Lipschitz properties for strongly local Dirichlet spaces VL - 281 ER - TY - JOUR AB - 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. AU - Lenz, Daniel AU - Weinmann, Timon AU - Wirth, Melchior ID - 9627 IS - 3 JF - Proceedings of the Edinburgh Mathematical Society SN - 0013-0915 TI - Self-adjoint extensions of bipartite Hamiltonians VL - 64 ER - TY - THES AB - 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. AU - Portinale, Lorenzo ID - 10030 SN - 2663-337X TI - Discrete-to-continuum limits of transport problems and gradient flows in the space of measures ER - TY - GEN AB - 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. AU - Feliciangeli, Dario AU - Gerolin, Augusto AU - Portinale, Lorenzo ID - 9792 T2 - arXiv TI - A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature ER - TY - THES AB - 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. AU - Feliciangeli, Dario ID - 9733 SN - 2663-337X TI - The polaron at strong coupling ER - TY - JOUR AB - 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. AU - Carlen, Eric A. AU - Maas, Jan ID - 6358 IS - 2 JF - Journal of Statistical Physics SN - 00224715 TI - Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems VL - 178 ER - TY - CHAP AB - 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. AU - Akopyan, Arseniy AU - Karasev, Roman ED - Klartag, Bo'az ED - Milman, Emanuel ID - 74 SN - 00758434 T2 - Geometric Aspects of Functional Analysis TI - Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures VL - 2256 ER - TY - JOUR AB - 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. AU - Gerencser, Mate ID - 7388 IS - 3 JF - Annales de l'Institut Henri Poincaré C, Analyse non linéaire SN - 0294-1449 TI - Nondivergence form quasilinear heat equations driven by space-time white noise VL - 37 ER - TY - JOUR AB - 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. AU - Forkert, Dominik L ID - 7629 SN - 2663-337X TI - Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains ER - TY - JOUR AB - This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou–Benamou formula for the Kantorovich metric . Such metrics appear naturally in discretisations of -gradient flow formulations for dissipative PDE. However, it has recently been shown that these metrics do not in general converge to , unless strong geometric constraints are imposed on the discrete mesh. In this paper we prove that, in a 1-dimensional periodic setting, discrete transport metrics converge to a limiting transport metric with a non-trivial effective mobility. This mobility depends sensitively on the geometry of the mesh and on the non-local mobility at the discrete level. Our result quantifies to what extent discrete transport can make use of microstructure in the mesh to reduce the cost of transport. AU - Gladbach, Peter AU - Kopfer, Eva AU - Maas, Jan AU - Portinale, Lorenzo ID - 7573 IS - 7 JF - Journal de Mathematiques Pures et Appliquees SN - 00217824 TI - Homogenisation of one-dimensional discrete optimal transport VL - 139 ER - TY - GEN AB - We consider finite-volume approximations of Fokker-Planck equations on bounded convex domains in 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 Γ-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalising 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. AU - Forkert, Dominik L AU - Maas, Jan AU - Portinale, Lorenzo ID - 10022 T2 - arXiv TI - Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions ER - TY - JOUR AB - We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in ℝd. These metrics are natural discrete counterparts to the Kantorovich metric 𝕎2, defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric Wt in terms of 𝕎2, as the size of the mesh T tends to 0. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov-Hausdorff convergence of the transport metric. AU - Gladbach, Peter AU - Kopfer, Eva AU - Maas, Jan ID - 71 IS - 3 JF - SIAM Journal on Mathematical Analysis SN - 00361410 TI - Scaling limits of discrete optimal transport VL - 52 ER - TY - JOUR AB - The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of α-Hölder drift in the recent literature the rate α/2 was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to 1/2 for all α>0. The result extends to Dini continuous coefficients, while in d=1 also to all bounded measurable coefficients. AU - Dareiotis, Konstantinos AU - Gerencser, Mate ID - 6359 JF - Electronic Journal of Probability TI - On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift VL - 25 ER - TY - JOUR AB - We consider the symmetric simple exclusion process in Zd with quenched bounded dynamic random conductances and prove its hydrodynamic limit in path space. The main tool is the connection, due to the self-duality of the process, between the invariance principle for single particles starting from all points and the macroscopic behavior of the density field. While the hydrodynamic limit at fixed macroscopic times is obtained via a generalization to the time-inhomogeneous context of the strategy introduced in [41], in order to prove tightness for the sequence of empirical density fields we develop a new criterion based on the notion of uniform conditional stochastic continuity, following [50]. In conclusion, we show that uniform elliptic dynamic conductances provide an example of environments in which the so-called arbitrary starting point invariance principle may be derived from the invariance principle of a single particle starting from the origin. Therefore, our hydrodynamics result applies to the examples of quenched environments considered in, e.g., [1], [3], [6] in combination with the hypothesis of uniform ellipticity. AU - Redig, Frank AU - Saada, Ellen AU - Sau, Federico ID - 8973 JF - Electronic Journal of Probability TI - Symmetric simple exclusion process in dynamic environment: Hydrodynamics VL - 25 ER - TY - JOUR AB - We consider an optimal control problem for an abstract nonlinear dissipative evolution equation. The differential constraint is penalized by augmenting the target functional by a nonnegative global-in-time functional which is null-minimized in the evolution equation is satisfied. Different variational settings are presented, leading to the convergence of the penalization method for gradient flows, noncyclic and semimonotone flows, doubly nonlinear evolutions, and GENERIC systems. AU - Portinale, Lorenzo AU - Stefanelli, Ulisse ID - 7550 IS - 2 JF - Advances in Mathematical Sciences and Applications SN - 1343-4373 TI - Penalization via global functionals of optimal-control problems for dissipative evolution VL - 28 ER - TY - JOUR AB - A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that appear in the usual formulation are not even defined in the Itô sense. AU - Gerencser, Mate AU - Gyöngy, István ID - 301 IS - 3 JF - Stochastic Processes and their Applications TI - A Feynman–Kac formula for stochastic Dirichlet problems VL - 129 ER - TY - JOUR AB - We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well-posedness and L1-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators Δ(|u|m−1u) for all m∈(1,∞), and Hölder continuous diffusion nonlinearity with exponent 1/2. AU - Dareiotis, Konstantinos AU - Gerencser, Mate AU - Gess, Benjamin ID - 65 IS - 6 JF - Journal of Differential Equations TI - Entropy solutions for stochastic porous media equations VL - 266 ER - TY - JOUR AB - We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories for singular stochastic PDEs with boundary conditions. The calculus of modelled distributions established in Hairer (Invent Math 198(2):269–504, 2014. https://doi.org/10.1007/s00222-014-0505-4) is extended to this setting. We formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the Dirichlet and Neumann heat kernels. These results are then used to provide solution theories for the KPZ equation with Dirichlet and Neumann boundary conditions and for the 2D generalised parabolic Anderson model with Dirichlet boundary conditions. In the case of the KPZ equation with Neumann boundary conditions, we show that, depending on the class of mollifiers one considers, a “boundary renormalisation” takes place. In other words, there are situations in which a certain boundary condition is applied to an approximation to the KPZ equation, but the limiting process is the Hopf–Cole solution to the KPZ equation with a different boundary condition. AU - Gerencser, Mate AU - Hairer, Martin ID - 319 IS - 3-4 JF - Probability Theory and Related Fields SN - 01788051 TI - Singular SPDEs in domains with boundaries VL - 173 ER - TY - JOUR AB - We give a construction allowing us to build local renormalized solutions to general quasilinear stochastic PDEs within the theory of regularity structures, thus greatly generalizing the recent results of [1, 5, 11]. Loosely speaking, our construction covers quasilinear variants of all classes of equations for which the general construction of [3, 4, 7] applies, including in particular one‐dimensional systems with KPZ‐type nonlinearities driven by space‐time white noise. In a less singular and more specific case, we furthermore show that the counterterms introduced by the renormalization procedure are given by local functionals of the solution. The main feature of our construction is that it allows exploitation of a number of existing results developed for the semilinear case, so that the number of additional arguments it requires is relatively small. AU - Gerencser, Mate AU - Hairer, Martin ID - 6028 IS - 9 JF - Communications on Pure and Applied Mathematics TI - A solution theory for quasilinear singular SPDEs VL - 72 ER - TY - JOUR AB - The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly—and in a sense, arbitrarily—bad: as shown by Krylov[ SIAM J. Math. Anal.34(2003) 1167–1182], for any α>0 one can find a simple 1-dimensional constant coefficient linear equation whose solution at the boundary is not α-Hölder continuous.We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on C1 domains are proved to be α-Hölder continuous up to the boundary with some α>0. AU - Gerencser, Mate ID - 6232 IS - 2 JF - Annals of Probability SN - 00911798 TI - Boundary regularity of stochastic PDEs VL - 47 ER - TY - JOUR AB - Starting from a microscopic model for a system of neurons evolving in time which individually follow a stochastic integrate-and-fire type model, we study a mean-field limit of the system. Our model is described by a system of SDEs with discontinuous coefficients for the action potential of each neuron and takes into account the (random) spatial configuration of neurons allowing the interaction to depend on it. In the limit as the number of particles tends to infinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, with derivatives only with respect to one variable and discontinuous coefficients. We also study strong well-posedness of the system of SDEs and prove the existence and uniqueness of a weak measure-valued solution to the PDE, obtained as the limit of the laws of the empirical measures for the system of particles. AU - Flandoli, Franco AU - Priola, Enrico AU - Zanco, Giovanni A ID - 10878 IS - 6 JF - Discrete and Continuous Dynamical Systems KW - Applied Mathematics KW - Discrete Mathematics and Combinatorics KW - Analysis SN - 1553-5231 TI - A mean-field model with discontinuous coefficients for neurons with spatial interaction VL - 39 ER - TY - JOUR AB - We consider the space of probability measures on a discrete set X, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset Y⊆X, it is natural to ask whether they can be connected by a constant speed geodesic with support in Y at all times. Our main result answers this question affirmatively, under a suitable geometric condition on Y introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi equations, which is of independent interest. AU - Erbar, Matthias AU - Maas, Jan AU - Wirth, Melchior ID - 73 IS - 1 JF - Calculus of Variations and Partial Differential Equations SN - 09442669 TI - On the geometry of geodesics in discrete optimal transport VL - 58 ER - TY - JOUR AB - We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition having density ρ on ℤ− and λ on ℤ+, and a second class particle initially at the origin. For ρ<λ, there is a shock and the second class particle moves with speed 1−λ−ρ. For large time t, we show that the position of the second class particle fluctuates on a t1/3 scale and determine its limiting law. We also obtain the limiting distribution of the number of steps made by the second class particle until time t. AU - Ferrari, Patrick AU - Ghosal, Promit AU - Nejjar, Peter ID - 72 IS - 3 JF - Annales de l'institut Henri Poincare (B) Probability and Statistics SN - 0246-0203 TI - Limit law of a second class particle in TASEP with non-random initial condition VL - 55 ER - TY - JOUR AB - Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces having a product structure with the noise in a Hilbert component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent Itô calculus. AU - Flandoli, Franco AU - Russo, Francesco AU - Zanco, Giovanni A ID - 1215 IS - 2 JF - Journal of Theoretical Probability TI - Infinite-dimensional calculus under weak spatial regularity of the processes VL - 31 ER - TY - JOUR AB - We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle. AU - Akopyan, Arseniy AU - Avvakumov, Sergey ID - 6355 JF - Forum of Mathematics, Sigma SN - 2050-5094 TI - Any cyclic quadrilateral can be inscribed in any closed convex smooth curve VL - 6 ER - TY - JOUR AB - We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by a≥0, which creates a shock in the particle density of order aT−1/3, T the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit lima→∞limT→∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order 1. This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several airy processes. AU - Nejjar, Peter ID - 70 IS - 2 JF - Latin American Journal of Probability and Mathematical Statistics SN - 1980-0436 TI - Transition to shocks in TASEP and decoupling of last passage times VL - 15 ER - TY - GEN AB - We prove that any convex body in the plane can be partitioned into m convex parts of equal areas and perimeters for any integer m≥2; this result was previously known for prime powers m=pk. We also give a higher-dimensional generalization. AU - Akopyan, Arseniy AU - Avvakumov, Sergey AU - Karasev, Roman ID - 75 TI - Convex fair partitions into arbitrary number of pieces ER - TY - JOUR AB - We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of “free boundary states.” For the case of one free boundary, our approach yields a new proof that the process is pfaffian. For the case of two free boundaries, we find that the process is not pfaffian, but a closely related process is. We also study three different applications of the Schur process with one free boundary: fluctuations of symmetrized last passage percolation models, limit shapes and processes for symmetric plane partitions and for plane overpartitions. AU - Betea, Dan AU - Bouttier, Jeremie AU - Nejjar, Peter AU - Vuletic, Mirjana ID - 556 IS - 12 JF - Annales Henri Poincare SN - 1424-0637 TI - The free boundary Schur process and applications I VL - 19 ER - TY - JOUR AB - In a recent article (Jentzen et al. 2016 Commun. Math. Sci. 14, 1477–1500 (doi:10.4310/CMS.2016.v14. n6.a1)), it has been established that, for every arbitrarily slow convergence speed and every natural number d ? {4, 5, . . .}, there exist d-dimensional stochastic differential equations with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence. In this paper, we strengthen the above result by proving that this slow convergence phenomenon also arises in two (d = 2) and three (d = 3) space dimensions. AU - Gerencser, Mate AU - Jentzen, Arnulf AU - Salimova, Diyora ID - 560 IS - 2207 JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences SN - 13645021 TI - On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions VL - 473 ER - TY - JOUR AB - Cauchy problems with SPDEs on the whole space are localized to Cauchy problems on a ball of radius R. This localization reduces various kinds of spatial approximation schemes to finite dimensional problems. The error is shown to be exponentially small. As an application, a numerical scheme is presented which combines the localization and the space and time discretization, and thus is fully implementable. AU - Gerencser, Mate AU - Gyöngy, István ID - 642 IS - 307 JF - Mathematics of Computation SN - 00255718 TI - Localization errors in solving stochastic partial differential equations in the whole space VL - 86 ER - TY - CHAP AB - We give a short overview on a recently developed notion of Ricci curvature for discrete spaces. This notion relies on geodesic convexity properties of the relative entropy along geodesics in the space of probability densities, for a metric which is similar to (but different from) the 2-Wasserstein metric. The theory can be considered as a discrete counterpart to the theory of Ricci curvature for geodesic measure spaces developed by Lott–Sturm–Villani. AU - Maas, Jan ED - Najman, Laurent ED - Romon, Pascal ID - 649 SN - 978-3-319-58001-2 T2 - Modern Approaches to Discrete Curvature TI - Entropic Ricci curvature for discrete spaces VL - 2184 ER - TY - CONF AB - We present a generalized optimal transport model in which the mass-preserving constraint for the L2-Wasserstein distance is relaxed by introducing a source term in the continuity equation. The source term is also incorporated in the path energy by means of its squared L2-norm in time of a functional with linear growth in space. This extension of the original transport model enables local density modulations, which is a desirable feature in applications such as image warping and blending. A key advantage of the use of a functional with linear growth in space is that it allows for singular sources and sinks, which can be supported on points or lines. On a technical level, the L2-norm in time ensures a disintegration of the source in time, which we use to obtain the well-posedness of the model and the existence of geodesic paths. The numerical discretization is based on the proximal splitting approach [18] and selected numerical test cases show the potential of the proposed approach. Furthermore, the approach is applied to the warping and blending of textures. AU - Maas, Jan AU - Rumpf, Martin AU - Simon, Stefan ED - Lauze, François ED - Dong, Yiqiu ED - Bjorholm Dahl, Anders ID - 989 SN - 03029743 TI - Transport based image morphing with intensity modulation VL - 10302 ER - TY - JOUR AB - We study a class of ergodic quantum Markov semigroups on finite-dimensional unital C⁎-algebras. These semigroups have a unique stationary state σ, and we are concerned with those that satisfy a quantum detailed balance condition with respect to σ. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to σ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the 2-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical 2-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance. AU - Carlen, Eric AU - Maas, Jan ID - 956 IS - 5 JF - Journal of Functional Analysis SN - 00221236 TI - Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance VL - 273 ER - TY - JOUR AB - We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied in Ferrari and Pimentel (2005a) for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deter- ministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of Ferrari and Nejjar (2015). AU - Ferrari, Patrik AU - Nejjar, Peter ID - 447 JF - Revista Latino-Americana de Probabilidade e Estatística TI - Fluctuations of the competition interface in presence of shocks VL - 9 ER - TY - JOUR AB - We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the d-dimensional cube, for arbitrary . The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker-Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution. AU - Maas, Jan AU - Matthes, Daniel ID - 1261 IS - 7 JF - Nonlinearity TI - Long-time behavior of a finite volume discretization for a fourth order diffusion equation VL - 29 ER - TY - JOUR AB - We develop a new and systematic method for proving entropic Ricci curvature lower bounds for Markov chains on discrete sets. Using different methods, such bounds have recently been obtained in several examples (e.g., 1-dimensional birth and death chains, product chains, Bernoulli–Laplace models, and random transposition models). However, a general method to obtain discrete Ricci bounds had been lacking. Our method covers all of the examples above. In addition we obtain new Ricci curvature bounds for zero-range processes on the complete graph. The method is inspired by recent work of Caputo, Dai Pra and Posta on discrete functional inequalities. AU - Fathi, Max AU - Maas, Jan ID - 1448 IS - 3 JF - The Annals of Applied Probability TI - Entropic Ricci curvature bounds for discrete interacting systems VL - 26 ER - TY - JOUR AB - We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer that this functional is asymptotically equivalent (in the sense of Γ-convergence) to the Jordan-Kinderlehrer-Otto functional arising in the Wasserstein gradient flow structure of the Fokker-Planck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof of Duong et al relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of Adams et al to arbitrary dimensions. AU - Erbar, Matthias AU - Maas, Jan AU - Renger, Michiel ID - 1517 JF - Electronic Communications in Probability TI - From large deviations to Wasserstein gradient flows in multiple dimensions VL - 20 ER - TY - JOUR AB - In this paper the optimal transport and the metamorphosis perspectives are combined. For a pair of given input images geodesic paths in the space of images are defined as minimizers of a resulting path energy. To this end, the underlying Riemannian metric measures the rate of transport cost and the rate of viscous dissipation. Furthermore, the model is capable to deal with strongly varying image contrast and explicitly allows for sources and sinks in the transport equations which are incorporated in the metric related to the metamorphosis approach by Trouvé and Younes. In the non-viscous case with source term existence of geodesic paths is proven in the space of measures. The proposed model is explored on the range from merely optimal transport to strongly dissipative dynamics. For this model a robust and effective variational time discretization of geodesic paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals. These functionals are defined on corresponding pairs of intensity functions and on associated pairwise matching deformations. Existence of time discrete geodesics is demonstrated. Furthermore, a finite element implementation is proposed and applied to instructive test cases and to real images. In the non-viscous case this is compared to the algorithm proposed by Benamou and Brenier including a discretization of the source term. Finally, the model is generalized to define discrete weighted barycentres with applications to textures and objects. AU - Maas, Jan AU - Rumpf, Martin AU - Schönlieb, Carola AU - Simon, Stefan ID - 1639 IS - 6 JF - ESAIM: Mathematical Modelling and Numerical Analysis TI - A generalized model for optimal transport of images including dissipation and density modulation VL - 49 ER - TY - JOUR AB - We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the n-dimensional discrete cube (the so-called Bernoulli-Laplace model) and the random transposition shuffle of the symmetric group of permutations on n letters. AU - Erbar, Matthias AU - Maas, Jan AU - Tetali, Prasad ID - 1635 IS - 4 JF - Annales de la faculté des sciences de Toulouse TI - Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition models VL - 24 ER -