Curvature-dimension Condition Research Articles

Porous media equation on locally finite graphs

In this paper, we consider two typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. The other one is to obtain global nontrivial nonnegative solution to porous-media equation via the use of Aronson-Benilan argument. We use the curvature dimension condition to give a characterization two point graph. We also give a porous-media equation criterion about stochastic completeness of the graph. There is not much work in the direction of the study of nonlinear heat equations on locally finite connected graphs.

Sobolev-to-Lipschitz property on {mathsf {QCD}}-spaces and applications

We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature-dimension condition recently introduced in Milman (Commun Pure Appl Math, to appear). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat semigroup. These results apply in particular to large classes of (ideal) sub-Riemannian manifolds.

Leaves decompositions in Euclidean spaces

We partly extend the localisation technique from convex geometry to the multiple constraints setting.For a given 1-Lipschitz map u:Rn→Rm, m≤n, we define and prove the existence of a partition of Rn, up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of u is an isometry on these sets.We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension m, the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag.

Curvature-dimension conditions under time change

We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry–Émery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension condition in the sense of Lott–Sturm–Villani will transform under time change.

The globalization theorem for the Curvature-Dimension condition

The Lott–Sturm–Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that it is enough to verify this condition locally: an essentially non-branching metric-measure space (X,mathsf {d},{mathfrak {m}}) (so that (text {supp}({mathfrak {m}}),mathsf {d}) is a length-space and {mathfrak {m}}(X) < infty ) verifying the local Curvature-Dimension condition {mathsf {CD}}_{loc}(K,N) with parameters K in {mathbb {R}} and N in (1,infty ), also verifies the global Curvature-Dimension condition {mathsf {CD}}(K,N). In other words, the Curvature-Dimension condition enjoys the globalization (or local-to-global) property, answering a question which had remained open since the beginning of the theory. For the proof, we establish an equivalence between L^1- and L^2-optimal-transport–based interpolation. The challenge is not merely a technical one, and several new conceptual ingredients which are of independent interest are developed: an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order temporal derivative of associated Kantorovich potentials; a surprising third-order theory for the latter Kantorovich potentials, which holds in complete generality on any proper geodesic space; and a certain rigidity property of the change-of-variables formula, allowing us to bootstrap the a-priori available regularity. As a consequence, numerous variants of the Curvature-Dimension condition proposed by various authors throughout the years are shown to, in fact, all be equivalent in the above setting, thereby unifying the theory.

Displacement convexity of Entropy and the distance cost Optimal Transportation

During the last decade Optimal Transport had a relevant role in the study of geometry of singular spaces that culminated with the Lott–Sturm–Villani theory. The latter is built on the characterisation of Ricci curvature lower bounds in terms of displacement convexity of certain entropy functionals along W 2 -geodesics. Substantial recent advancements in the theory (localization paradigm and local-to-global property) have been obtained considering the different point of view of L 1 -Optimal transport problems yielding a different curvature dimension CD 1 (K,N) [5] formulated in terms of one-dimensional curvature properties of integral curves of Lipschitz maps. In this note we show that the two approaches produce the same curvature-dimension condition reconciling the two definitions. In particular we show that the CD 1 (K,N) condition can be formulated in terms of displacement convexity along W 1 -geodesics.

Stability of metric measure spaces with integral Ricci curvature bounds

In this article we study stability and compactness w.r.t. measured Gromov-Hausdorff convergence of smooth metric measure spaces with integral Ricci curvature bounds. More precisely, we prove that a sequence of n-dimensional Riemannian manifolds subconverges to a metric measure space that satisfies the curvature-dimension condition CD(K,n) in the sense of Lott-Sturm-Villani provided the Lp-norm for p>n2 of the part of the Ricci curvature that lies below K converges to 0. The results also hold for sequences of general smooth metric measure spaces (M,gM,e−fvolM) where Bakry-Emery curvature replaces Ricci curvature. Corollaries are a Brunn-Minkowski-type inequality, a Bonnet-Myers estimate and a statement on finiteness of the fundamental group. Together with a uniform noncollapsing condition the limit even satisfies the Riemannian curvature-dimension condition RCD(K,N). This implies volume and diameter almost rigidity theorems.

Stratified spaces and synthetic Ricci curvature bounds

We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K ∈ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2π and its dimension is at most equal to N. This gives a new wide class of geometric examples of metric measure spaces satisfying the RCD(K, N) curvature-dimension condition, including for instance spherical suspensions, orbifolds, Kahler-Einstein manifolds with a divisor, Einstein manifolds with conical singularities along a curve. We also obtain new analytic and geometric results on stratied spaces, such as Bishop-Gromov volume inequality, Laplacian comparison, Levy-Gromov isoperimetric inequality.

Blow-up for a semilinear heat equation with Fujita’s critical exponent on locally finite graphs

Let \(G=(V,E)\) be a locally finite, connected and weighted graph. We prove that, for a graph satisfying curvature dimension condition \(CDE'(n,0)\) and uniform polynomial volume growth of degree m, all non-negative solutions of the equation \(\partial _tu=\Delta u+u^{1+\alpha }\) blow up in a finite time, provided that \(\alpha =\frac{2}{m}\). We also consider the blow-up problem under certain conditions for volume growth and initial value. These results complement our previous work joined with Lin.

Independence of synthetic curvature dimension conditions on transport distance exponent

Independence of synthetic curvature dimension conditions on transport distance exponent

The entropy method under curvature-dimension conditions in the spirit of Bakry-Émery in the discrete setting of Markov chains

We consider continuous-time (not necessarily finite) Markov chains on discrete spaces and identify a curvature-dimension inequality, the condition CDϒ(κ,∞), which serves as a natural analogue of the classical Bakry-Émery condition CD(κ,∞) in several respects. In particular, it is tailor-made to the classical approach of proofing the modified logarithmic Sobolev inequality via computing and estimating the second time derivative of the entropy along the heat flow generated by the generator of the Markov chain. We prove that curvature bounds in the sense of CDϒ are preserved under tensorization, discuss links to other notions of discrete curvature and consider a variety of examples including complete graphs, the hypercube and birth-death processes. We further consider power type entropies and determine, in the same spirit, a natural CD condition which leads to Beckner inequalities. The CDϒ condition is also shown to be compatible with the diffusive setting, in the sense that corresponding hybrid processes enjoy a tensorization property.

Curvature-Dimension Condition Meets Gromov's n-Volumic Scalar Curvature

We study the properties of the $n$-volumic scalar curvature in this note. Lott-Sturm-Villani's curvature-dimension condition ${\rm CD}(\kappa,n)$ was showed to imply Gromov's $n$-volumic scalar curvature $\geq n\kappa$ under an additional $n$-dimensional condition and we show the stability of $n$-volumic scalar curvature $\geq \kappa$ with respect to smGH-convergence. Then we propose a new weighted scalar curvature on the weighted Riemannian manifold and show its properties.

Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains

In the setting of reversible continuous-time Markov chains, the CDΥ condition has been shown recently to be a consistent analogue to the Bakry-Émery condition in the diffusive setting in terms of proving Li-Yau inequalities under a finite dimension term and proving the modified logarithmic Sobolev inequality under a positive curvature bound. In this article we examine the case where both is given, a finite dimension term and a positive curvature bound. For this purpose we introduce the CDΥ(κ,F) condition, where the dimension term is expressed by a so called CD-function F. We derive functional inequalities relating the entropy to the Fisher information, which we will call entropy-information inequalities. Further, we deduce applications of entropy-information inequalities such as ultracontractivity bounds, exponential integrability of Lipschitz functions, finite diameter bounds and a modified version of the celebrated Nash inequality.

Dynamical aspects of the generalized Schrödinger problem via Otto calculus – A heuristic point of view

The defining equation (\ast)\qquad \dot \omega_t=-F'(\omega_t) of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation (\ast) into the family of slowed down gradient flow equations: \dot \omega^{\varepsilon}_t=- \varepsilon F'( \omega ^{ \varepsilon}_t) , where \varepsilon &gt; 0 , and (ii) by considering the accelerations \ddot \omega ^{ \varepsilon}_t . We shall focus on Wasserstein gradient flows. Our approach is mainly heuristic. It relies on Otto calculus. A special formulation of the Schrödinger problem consists in minimizing some action on the Wasserstein space of probability measures on a Riemannian manifold subject to fixed initial and final data. We extend this action minimization problem by replacing the usual entropy, underlying the Schrödinger problem, with a general function on the Wasserstein space. The corresponding minimal cost approaches the squared Wasserstein distance when the fluctuation parameter \varepsilon tends to zero. We show heuristically that the solutions satisfy some Newton equation, extending a recent result of Conforti. The connection with Wasserstein gradient flows is established and various inequalities, including evolutional variational inequalities and contraction inequalities under a curvature-dimension condition, are derived with a heuristic point of view. As a rigorous result we prove a new and general contraction inequality for the Schrödinger problem under a Ricci lower bound on a smooth and compact Riemannian manifold.

The heat flow on metric random walk spaces

In this paper we study the Heat Flow on Metric Random Walk Spaces, which unifies into a broad framework the heat flow on locally finite weighted connected graphs, the heat flow determined by finite Markov chains and some nonlocal evolution problems. We give different characterizations of the ergodicity and prove that a metric random walk space with positive Ollivier-Ricci curvature is ergodic. Furthermore, we prove a Cheeger inequality and, as a consequence, we show that a Poincaré inequality holds if, and only if, an isoperimetric inequality holds. We also study the Bakry-Émery curvature-dimension condition and its relation with functional inequalities like the Poincaré inequality and the transport-information inequalities.

Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces

Aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X, d,m). On the geometric side, our new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N -dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, our new approach uses the nonlinear diffusion semigroup induced by the N -dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger’s energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm.

Stabilities of rough curvature dimension condition

We study the asymptotic behavior of metric measure spaces satisfying the rough curvature dimension condition. We prove stabilities of the rough curvature dimension condition with respect to the observable distance function and the $L^2$-transportation distance function.

Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions

We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the d-dimensional torus with singular p-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with divergence-free coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative Itô-formula.

Regularity of Lagrangian flows over RCD*(K, N) spaces

Abstract The aim of this note is to provide regularity results for Regular Lagrangian flows of Sobolev vector fields over compact metric measure spaces verifying the Riemannian curvature dimension condition. We first prove, borrowing some ideas already present in the literature, that flows generated by vector fields with bounded symmetric derivative are Lipschitz, providing the natural extension of the standard Cauchy–Lipschitz theorem to this setting. Then we prove a Lusin-type regularity result in the Sobolev case (under the additional assumption that the m.m.s. is Ahlfors regular) therefore extending the already known Euclidean result.

Stability of RCD condition under concentration topology

We prove the stability of the Riemannian curvature dimension condition introduced by Ambrosio–Gigli–Savare under the concentration of metric measure spaces introduced by Gromov. This is an analogue of the result of Funano–Shioya for the curvature dimension condition of Lott–Villani and Sturm. These conditions are synthetic lower Ricci curvature bound for metric measure spaces. En route, we also prove the convergence of the Cheeger energy in our setting.