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Jan Maas

Publications and source records attributed to Jan Maas.

At least 37 records · Page 2Linked to original sources
arXiv2016

Entropic Ricci curvature bounds for discrete interacting systems

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.

math.PR↗
arXiv2016

From large deviations to Wasserstein gradient flows in multiple dimensions

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 in \cite{DLR2013} 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 \cite{ADPZ2011} to arbitrary dimensions.

math.PR↗
arXiv2015

Long-time behavior of a finite volume discretization for a fourth order diffusion equation

We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the $d$-dimensional cube, for arbitrary $d \geq 1$. 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.

math.AP↗
arXiv2015

A generalized model for optimal transport of images including dissipation and density modulation

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.

math.NA↗
arXiv2013

Poisson stochastic integration in Banach spaces

We prove new upper and lower bounds for Banach space-valued stochastic integrals with respect to a compensated Poisson random measure. Our estimates apply to Banach spaces with non-trivial martingale (co)type and extend various results in the literature. We also develop a Malliavin framework to interpret Poisson stochastic integrals as vector-valued Skorohod integrals, and prove a Clark-Ocone representation formula.

math.PR↗
arXiv2013

Approximating rough stochastic PDEs

We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently in [Hairer, Weber, Probab. Theory Related Fields, to appear]. The key idea was to use the theory of controlled rough paths to give definitions of weak / mild solutions and to set up a Picard iteration argument. In this article the limiting behaviour of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Itô-Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.

math.PR↗
arXiv2013

Gromov-Hausdorff convergence of discrete transportation metrics

This paper continues the investigation of `Wasserstein-like' transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics on the d-dimensional discrete torus with mesh size 1/N converge, when $N\to\infty$, to the standard 2-Wasserstein distance W_2 on the continuous torus in the sense of Gromov-Hausdorff. This is the first convergence result for the recently developed discrete transportation metrics. The result shows the compatibility between these metrics and the well-established 2-Wasserstein metric.

math.MG↗
arXiv2012

Gradient flow structures for discrete porous medium equations

We consider discrete porous medium equations of the form \partial_t ρ_t = Δϕ(ρ_t), where Δis the generator of a reversible continuous time Markov chain on a finite set X, and ϕis an increasing function. We show that these equations arise as gradient flows of certain entropy functionals with respect to suitable non-local transportation metrics. This may be seen as a discrete analogue of the Wasserstein gradient flow structure for porous medium equations in R^n discovered by Otto. We present a one-dimensional counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence to the Wasserstein metric.

math.FA↗
arXiv2012

A spatial version of the Itô-Stratonovich correction

We consider a class of stochastic PDEs of Burgers type in spatial dimension 1, driven by space-time white noise. Even though it is well known that these equations are well posed, it turns out that if one performs a spatial discretization of the nonlinearity in the "wrong" way, then the sequence of approximate equations does converge to a limit, but this limit exhibits an additional correction term. This correction term is proportional to the local quadratic cross-variation (in space) of the gradient of the conserved quantity with the solution itself. This can be understood as a consequence of the fact that for any fixed time, the law of the solution is locally equivalent to Wiener measure, where space plays the role of time. In this sense, the correction term is similar to the usual Itô-Stratonovich correction term that arises when one considers different temporal discretizations of stochastic ODEs.

math.PR↗
arXiv2012

Ricci curvature of finite Markov chains via convexity of the entropy

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry--Emery and Otto--Villani. Furthermore we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.

math.MG↗
arXiv2012

An Analog of the 2-Wasserstein Metric in Non-commutative Probability under which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy

Let $\Cl$ denote the Clifford algebra over $\R^n$, which is the von Neumann algebra generated by $n$ self-adjoint operators $Q_j$, $j=1,...,n$ satisfying the canonical anticommutation relations, $Q_iQ_j+Q_jQ_i = 2δ_{ij}I$, and let $τ$ denote the normalized trace on $\Cl$. This algebra arises in quantum mechanics as the algebra of observables generated by $n$ Fermionic degrees of freedom. Let $\Dens$ denote the set of all positive operators $ρ\in\Cl$ such that $τ(ρ) =1$; these are the non-commutative analogs of probability densities in the non-commutative probability space $(\Cl,τ)$. The Fermionic Fokker-Planck equation is a quantum-mechanical analog of the classical Fokker-Planck equation with which it has much in common, such as the same optimal hypercontractivity properties. In this paper we construct a Riemannian metric on $\Dens$ that we show to be a natural analog of the classical 2-Wasserstein metric, and we show that, in analogy with the classical case, the Fermionic Fokker-Planck equation is gradient flow in this metric for the relative entropy with respect to the ground state. We derive a number of consequences of this, such as a sharp Talagrand inequality for this metric, and we prove a number of results pertaining to this metric. Several open problems are raised.

math.FA↗
arXiv2011

Gradient flows of the entropy for finite Markov chains

Let K be an irreducible and reversible Markov kernel on a finite set X. We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wasserstein gradient flow interpretation of the heat flow in R^n by Jordan, Kinderlehrer, and Otto (1998). The metric W is similar to, but different from, the L^2-Wasserstein metric, and is defined via a discrete variant of the Benamou-Brenier formula.

math.PR↗
arXiv2011

Gradient estimates and domain identification for analytic Ornstein-Uhlenbeck operators

Let (P(t)) be the Ornstein-Uhlenbeck semigroup associated with the stochastic Cauchy problem dU(t) = AU(t)dt + dW_H(t), where A is the generator of a C_0-semigroup (S(t)) on a Banach space E, H is a Hilbert subspace of E, and (W_H(t)) is an H-cylindrical Brownian motion. Assuming that (S(t)) restricts to a C_0-semigroup on H, we obtain L^p-bounds for the gradient D_H P(t). We show that if (P(t)) is analytic, then the invariance assumption is fulfilled. As an application we determine the L^p-domain of the generator of (P(t)) explicitly in the case where (S(t)) restricts to a C_0-semigroup on H which is similar to an analytic contraction semigroup. The results are applied to the 1D stochastic heat equation driven by additive space-time white noise.

math.FA↗
arXiv2010

Non-tangential maximal functions and conical square functions with respect to the Gaussian measure

We study, in $L^{1}(\R^n;γ)$ with respect to the gaussian measure, non-tangential maximal functions and conical square functions associated with the Ornstein-Uhlenbeck operator by developing a set of techniques which allow us, to some extent, to compensate for the non-doubling character of the gaussian measure. The main result asserts that conical square functions can be controlled in $L^1$-norm by non-tangential maximal functions. Along the way we prove a change of aperture result for the latter. This complements recent results on gaussian Hardy spaces due to Mauceri and Meda.

math.FA↗
arXiv2010

A Trotter product formula for gradient flows in metric spaces

We prove a Trotter product formula for gradient flows in metric spaces. This result is applied to establish convergence in the L^2-Wasserstein metric of the splitting method for some Fokker-Planck equations and porous medium type equations perturbed by a potential.

math.AP↗
arXiv2008

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,mu) be an abstract Wiener space and let D_V := VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space G. Given a bounded operator B on G, coercive on the closure of the range of V, we consider the realisation of the operator D_V* B D_V in L^p(E,mu) for 1<p<\infty. Our main result states that the following assertions are equivalent: (1) dom((sqrt(D_V* B D_V)) = dom(D_V) and Meyer's inequalities hold for D_V* B D_V; (2) D_V D_V* B admits a bounded H-infinity calculus on the closure of the range of D_V; (3) dom(sqrt(V*BV)) = dom(V) and Meyer's inequalities hold for V*BV; (4) VV*B admits a bounded H-infinity calculus on the closure of the range of V. Moreover, if these conditions are satisfied, then dom(L) = dom(D_V^2) \cap dom(D_A). The equivalence of (1)-(4) is a non-symmetric generalisation of the classical Meyer inequalities (which correspond to the case G=H, V=I, B=I). A one-sided version of the main result, giving L^p-boundedness of the associated Riesz transforms in terms of a square function estimate, is also obtained. As an application let -A generate an analytic C_0-contraction semigroup on a Hilbert space H and let -L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A.

math.FA↗