Duality on gradient estimates and Wasserstein controls

2014

Duke Math. J.

442

688

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X, d, m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, globalto-local and local-to-global properties. In these spaces, that we call RCD(K, ∞) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-Emery estimates and the L ∞ − Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger's relaxed slope and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincaré and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.

Section: Stabilitymentioning

confidence: 91%

Metric measure spaces with Riemannian Ricci curvature bounded from below

Ambrosio¹,

Gigli²,

2014

Duke Math. J.

442

688

“…A few metric concepts are recalled in Section 3.1, whereas Section 3.2 shows how to construct a dual semigroup (H t ) t≥0 in the space of probability measures P(X) under suitable Lipschitz estimates on (P t ) t≥0 . By using refined properties of the Hopf-Lax semigroup, we also extend some of the duality results proved by Kuwada [34] to general complete and separable metric measure spaces, avoiding any doubling or Poincaré condition. Section 3.3 presents a careful analysis of the intrinsic distance d E (1.9) associated to a Dirichlet form and of Energy measure structures (X, τ, m, E).…”

mentioning

confidence: 84%

“…It is also worth mentioning that in this class of spaces BE(K, ∞) is equivalent to an (exponential) contraction property for the semigroup (H t ) t≥0 with respect to the Wasserstein distance W 2 (see Corollary 3.18), in analogy with [34].…”

mentioning

confidence: 99%

Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds

Ambrosio¹,

Gigli

2015

Ann. Probab.

223

377

The aim of the present paper is to bridge the gap between the Bakry-Émery and the Lott-Sturm-Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds.We start from a strongly local Dirichlet form E admitting a Carré du champ Γ in a Polish measure space (X, m) and a canonical distance dE that induces the original topology of X. We first characterize the distinguished class of Riemannian Energy measure spaces, where E coincides with the Cheeger energy induced by dE and where every function f with Γ(f ) ≤ 1 admits a continuous representative.In such a class, we show that if E satisfies a suitable weak form of the Bakry-Émery curvature dimension condition BE(K, ∞) then the metric measure space (X, d, m) satisfies the Riemannian Ricci curvature bound RCD(K, ∞) according to [Duke Math. J. 163 (2014) 1405-1490], thus showing the equivalence of the two notions.Two applications are then proved: the tensorization property for Riemannian Energy spaces satisfying the Bakry-Émery BE(K, N ) condition (and thus the corresponding one for RCD(K, ∞) spaces without assuming nonbranching) and the stability of BE(K, N ) with respect to Sturm-Gromov-Hausdorff convergence.

“…The first formulation relies on the Kuwada duality result [34], which exploits the dual dynamic representation formula (14a,b) and deals with a couple of dual maps: P : C b (X) → C b (X) linear and continuous and P * : P(X) → P(X) satisfying…”

Section: Pointwise Gradient Estimates and Wasserstein Contractionmentioning

confidence: 99%

Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space

Ambrosio¹,

Gigli²,

2017

EMS Newsl.

1Ricci curvature and Γ-calculus in a smooth settingIn order to introduce and explain the basic notions we will deal with, we consider a smooth, complete and connected ddimensional Riemannian manifold (M, g) endowed with its Riemannian distance d g and a reference measure m = e −V Vol g that is absolutely continuous with respect to the Riemannian volume form Vol g ; its density is associated to the smooth potential V : M → R. The metric tensor g induces a norm |∇ f | g of the gradient of a smooth function f : M → R, given in local coordinates by |∇ f | 2 g = i, j g i j ∂ i f ∂ j f . The combination with the reference measure m gives rise to the quadratic energy form (1) and to the second order differential operator L = ∆ g − ∇V, · g satisfying the integration-by-parts formulaand generates a semigroup (P t ) t≥0 through the solution f t = P t f of the diffusion equationWhenever f