Metric measure spaces with Riemannian Ricci curvature bounded from below

By Luigi Ambrosio, Nicola Gigli, Giuseppe Savar'e. 2011

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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, global-to-local and local-to-global properties. In these spaces, that we call RCD(K,\infty) 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^\infty-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 Poincar\'e and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/e60b188dd262abc380e91bb9659c77d3b1d28582

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Citations in the corpus

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below (2011, background): “eat ﬂows is complete. The paper ends, in the last section, with results that are important for the development of a “calculus” with Kantorovich potentials. They will play a key role in some proofs of [3]. We included these results here because their validity does not really depend on curvature prope”
Cheeger-harmonic functions in metric measure spaces revisited (2013, background): “Poincare´ inequality holds ifK ≥ 0. However, as CD(K,N) conditions include the Finsler geometry, it is not known if the heat semigroup curvature condition holds under them. Recently, Ambrosio et. al. [2] (see also [1]) introduced a Riemannian Ricci curvature condition RCD(K,∞) on metric spaces, unde”
The globalization theorem for the Curvature-Dimension condition (2016, background): “′,N(θ) in K′, the claim follows. 95 13.2 RCD(K,N) spaces We also mention the more recent Riemannian Curvature Dimension condition RCD∗(K,N). In the inﬁnite dimensional case N= ∞, it was introduced in [7] for ﬁnite measures mand in [4] for σ-ﬁnite ones. The class RCD∗(K,N) with N<∞ has been propos”
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, methodology): “r, the quasi Bakry-Emery curvature condition (´ 8) is indeed weaker than BE(0,∞). In the context of RCD spaces, the relation between the conditions RCD(0,∞) and BE(0,∞) is discussed in detail in [5], [10] and [37]. 3 BV class and co-area formula In this section we use the Dirichlet form and the asso”
On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, background): “Combining linearity of the heat flow with the CD(K,∞) condition leads to the Riemannian curvature condition RCD(K,∞) introduced in [4].”
On the volume measure of non-smooth spaces with Ricci curvature bounded below (2016, result): “Inspired by this fact, in [4], Ambrosio-Gigli-Savaré proposed a strengthening of the CD-condition in order to enforce, in some weak sense, a Riemannian-like behavior of spaces with a curvaturedimension bound (to be precise in [4] it was analyzed the case of strong-CD(K,∞) spaces endowed with a proba”
Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, background): “Let us recall from [5, 28] that, under this assumption, the function ∇f1 · ∇f2 := lim ε→0 |∇(f1 + εf2)|(2) − |∇f1|(2) 2ε defines a symmetric bilinear form on H(1,2)(X, d,m)×H1,2(X, d,m) with values into L(1)(X,m).”