@article{73,
abstract = {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.},
author = {Erbar, Matthias and Maas, Jan and Wirth, Melchior},
issn = {09442669},
journal = {Calculus of Variations and Partial Differential Equations},
number = {1},
publisher = {Springer},
title = {{On the geometry of geodesics in discrete optimal transport}},
doi = {10.1007/s00526-018-1456-1},
volume = {58},
year = {2019},
}
@article{6028,
abstract = {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.},
author = {Gerencser, Mate and Hairer, Martin},
journal = {Communications on Pure and Applied Mathematics},
publisher = {Wiley},
title = {{A solution theory for quasilinear singular SPDEs}},
doi = {10.1002/cpa.21816},
year = {2019},
}
@article{6232,
abstract = {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.},
author = {Gerencser, Mate},
issn = {00911798},
journal = {Annals of Probability},
number = {2},
pages = {804--834},
publisher = {Institute of Mathematical Statistics},
title = {{Boundary regularity of stochastic PDEs}},
doi = {10.1214/18-AOP1272},
volume = {47},
year = {2019},
}
@article{301,
abstract = {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.},
author = {Gerencser, Mate and Gyöngy, István},
journal = {Stochastic Processes and their Applications},
number = {3},
pages = {995--1012},
publisher = {Elsevier},
title = {{A Feynman–Kac formula for stochastic Dirichlet problems}},
doi = {10.1016/j.spa.2018.04.003},
volume = {129},
year = {2019},
}
@article{65,
abstract = {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.},
author = {Dareiotis, Konstantinos and Gerencser, Mate and Gess, Benjamin},
journal = {Journal of Differential Equations},
number = {6},
pages = {3732--3763},
publisher = {Elsevier},
title = {{Entropy solutions for stochastic porous media equations}},
doi = {10.1016/j.jde.2018.09.012},
volume = {266},
year = {2019},
}
@article{319,
abstract = {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.},
author = {Gerencser, Mate and Hairer, Martin},
journal = {Probability Theory and Related Fields},
publisher = {Springer},
title = {{Singular SPDEs in domains with boundaries}},
doi = {10.1007/s00440-018-0841-1},
year = {2018},
}
@unpublished{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 = {ArXiv},
pages = {15},
publisher = {ArXiv},
title = {{Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures}},
year = {2018},
}
@article{556,
abstract = {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.},
author = {Betea, Dan and Bouttier, Jeremie and Nejjar, Peter and Vuletic, Mirjana},
issn = {14240637},
journal = {Annales Henri Poincare},
number = {12},
pages = {3663--3742},
publisher = {Fakultät für Mathematik Universität Wien},
title = {{The free boundary Schur process and applications}},
doi = {10.1007/s00023-018-0723-1},
volume = {19},
year = {2018},
}
@unpublished{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},
publisher = {ArXiv},
title = {{Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems}},
year = {2018},
}
@unpublished{75,
abstract = {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.},
author = {Akopyan, Arseniy and Avvakumov, Sergey and Karasev, Roman},
booktitle = {ArXiv},
pages = {11},
publisher = {ArXiv},
title = {{Convex fair partitions into arbitrary number of pieces}},
year = {2018},
}
@article{70,
abstract = {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.},
author = {Nejjar, Peter},
issn = {1980-0436},
journal = {Latin American Journal of Probability and Mathematical Statistics},
number = {2},
pages = {1311--1334},
publisher = {ALEA},
title = {{Transition to shocks in TASEP and decoupling of last passage times}},
doi = {10.30757/ALEA.v15-49},
volume = {15},
year = {2018},
}
@article{1215,
abstract = {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.},
author = {Flandoli, Franco and Russo, Francesco and Zanco, Giovanni A},
journal = {Journal of Theoretical Probability},
number = {2},
pages = {789--826},
publisher = {Springer},
title = {{Infinite-dimensional calculus under weak spatial regularity of the processes}},
doi = {10.1007/s10959-016-0724-2},
volume = {31},
year = {2018},
}
@unpublished{6359,
abstract = {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 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 a class of everywhere discontinuous coefficients. },
author = {Dareiotis, Konstantinos and Gerencser, Mate},
pages = {12},
publisher = {ArXiv},
title = {{On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift}},
year = {2018},
}
@unpublished{71,
abstract = {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.},
author = {Gladbach, Peter and Kopfer, Eva and Maas, Jan},
booktitle = {ArXiv},
pages = {44},
publisher = {ArXiv},
title = {{Scaling limits of discrete optimal transport}},
year = {2018},
}
@article{6355,
abstract = {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.},
author = {Akopyan, Arseniy and Avvakumov, Sergey},
issn = {2050-5094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{Any cyclic quadrilateral can be inscribed in any closed convex smooth curve}},
doi = {10.1017/fms.2018.7},
volume = {6},
year = {2018},
}
@unpublished{72,
abstract = {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.},
author = {Ferrari, Patrick and Nejjar, Peter and Ghosal, Promit},
booktitle = {ArXiv},
publisher = {ArXiv},
title = {{Limit law of a second class particle in TASEP with non-random initial condition}},
year = {2018},
}
@inproceedings{989,
abstract = {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.},
author = {Maas, Jan and Rumpf, Martin and Simon, Stefan},
editor = {Lauze, François and Dong, Yiqiu and Bjorholm Dahl, Anders},
issn = {03029743},
location = {Kolding, Denmark},
pages = {563 -- 577},
publisher = {Springer},
title = {{Transport based image morphing with intensity modulation}},
doi = {10.1007/978-3-319-58771-4_45},
volume = {10302},
year = {2017},
}
@article{447,
abstract = {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).},
author = {Ferrari, Patrik and Nejjar, Peter},
journal = {Revista Latino-Americana de Probabilidade e Estatística},
pages = {299 -- 325},
publisher = {ALEA Network},
title = {{Fluctuations of the competition interface in presence of shocks}},
volume = {9},
year = {2017},
}
@inbook{649,
abstract = {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.},
author = {Maas, Jan},
booktitle = {Modern Approaches to Discrete Curvature},
editor = {Najman, Laurent and Romon, Pascal},
issn = {978-3-319-58002-9},
pages = {159 -- 174},
publisher = {Springer},
title = {{Entropic Ricci curvature for discrete spaces}},
doi = {10.1007/978-3-319-58002-9_5},
volume = {2184},
year = {2017},
}
@article{956,
abstract = {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.},
author = {Carlen, Eric and Maas, Jan},
issn = {00221236},
journal = {Journal of Functional Analysis},
number = {5},
pages = {1810 -- 1869},
publisher = {Academic Press},
title = {{Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance}},
doi = {10.1016/j.jfa.2017.05.003},
volume = {273},
year = {2017},
}
@article{560,
abstract = {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.},
author = {Gerencser, Mate and Jentzen, Arnulf and Salimova, Diyora},
issn = {13645021},
journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
number = {2207},
publisher = {Royal Society of London},
title = {{On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions}},
doi = {10.1098/rspa.2017.0104},
volume = {473},
year = {2017},
}
@article{642,
abstract = {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.},
author = {Gerencser, Mate and Gyöngy, István},
issn = {00255718},
journal = {Mathematics of Computation},
number = {307},
pages = {2373 -- 2397},
publisher = {American Mathematical Society},
title = {{Localization errors in solving stochastic partial differential equations in the whole space}},
doi = {10.1090/mcom/3201},
volume = {86},
year = {2017},
}
@article{1448,
abstract = {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.},
author = {Fathi, Max and Maas, Jan},
journal = {The Annals of Applied Probability},
number = {3},
pages = {1774 -- 1806},
publisher = {Institute of Mathematical Statistics},
title = {{Entropic Ricci curvature bounds for discrete interacting systems}},
doi = {10.1214/15-AAP1133},
volume = {26},
year = {2016},
}
@article{1261,
abstract = {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.},
author = {Maas, Jan and Matthes, Daniel},
journal = {Nonlinearity},
number = {7},
pages = {1992 -- 2023},
publisher = {IOP Publishing Ltd.},
title = {{Long-time behavior of a finite volume discretization for a fourth order diffusion equation}},
doi = {10.1088/0951-7715/29/7/1992},
volume = {29},
year = {2016},
}
@article{1635,
abstract = {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.},
author = {Erbar, Matthias and Maas, Jan and Tetali, Prasad},
journal = {Annales de la faculté des sciences de Toulouse},
number = {4},
pages = {781 -- 800},
publisher = {Univ. Paul Sabatier},
title = {{Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition models}},
doi = {10.5802/afst.1464},
volume = {24},
year = {2015},
}
@article{1517,
abstract = {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.
},
author = {Erbar, Matthias and Maas, Jan and Renger, Michiel},
journal = {Electronic Communications in Probability},
publisher = {Institute of Mathematical Statistics},
title = {{From large deviations to Wasserstein gradient flows in multiple dimensions}},
doi = {10.1214/ECP.v20-4315},
volume = {20},
year = {2015},
}
@article{1639,
abstract = {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.},
author = {Maas, Jan and Rumpf, Martin and Schönlieb, Carola and Simon, Stefan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
number = {6},
pages = {1745 -- 1769},
publisher = {EDP Sciences},
title = {{A generalized model for optimal transport of images including dissipation and density modulation}},
doi = {10.1051/m2an/2015043},
volume = {49},
year = {2015},
}