Duality on gradient estimates and Wasserstein controls

By Kazumasa Kuwada. 2009

143 citations recorded by the source corpus.

Abstract We establish a duality between L p -Wasserstein control and L q -gradient estimate in a general framework. Our result extends a known result for a heat flow on a Riemannian manifold. Especially, we can derive a Wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound. By applying our result to a subelliptic heat flow on a Lie group, we obtain a coupling of heat distributions which carries a good control of their relative distance.

Subfields: Applied Mathematics, Geometry and Topology, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/14c9fa4fdc224ce8f2df6961eca8fdf684e7103f

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In Applied Mathematics

In Geometry and Topology

In Radiology, Nuclear Medicine and Imaging

Citations in the corpus

On the differential structure of metric measure spaces and applications (2012, methodology): “This is achieved by calling into play optimal transport as key tool: we will combine gradient flow theory, a powerful duality principle due to Kuwada [26] and a general superposition principle due to Lisini [28], see the proof of Lemma 3.13.”
A PDE approach to a 2-dimensional matching problem (2016, methodology): “∞ Z D pt(x,y)dm(x) = [[pt(·,y)]]∞ ≤ Cuct−d/2. Writing µ= R Dδxdµ(x) and (DR) in the form W22(Pt∗δx,δx) ≤ Cdrt, from the joint convexity of W2 2 (2.5) we obtain W2 2 (P ∗ tµ,µ) ≤ Cdrt. By duality, see [K10], the gradient contractivity property leads to contractivity w.r.t. W2 distance W2 2 (P ∗ tµ,P ”
Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds (2012, methodology): “Then we shall prove, following the lines of [34], that in length metric spaces the pointwise Bakry-Émery-like assumption |DPtf |(2)(x) ≤ C(2)(t) P̃t|Df |(2)(x) for all x ∈ X , f ∈ Lipb(X) ∩ L(2)(X,m), (3.”
Geometry and Analysis of Dirichlet forms (2012, methodology): “In this section, we apply Corollary 2.1 to a variant of the dual formula of Kuwada [28] and the coarse Ricci curvature of Ollivier in Theorem 6.1 and Corollary 6.1, and then apply Theorem 5.1 to the Ricci curvatures of Bakry-Emery and Lott-Sturm-Villani in Corollary 6.2.”
On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, background): “[21] N. Gigli, K. Kuwada, and S.-I. Ohta.”
An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature (2014, background): “very general duality argument due to Kuwada [35].”
Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. (2011, methodology): “This lemma has been introduced in [17] and then used in [9, 2] to study the heat flow on metric measure spaces.”
Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces (2013, unknown): “suitably adapted to the weak integral version of (1.5). Finally, the application of (1.5) to contraction estimates for the heat ﬂow (Ht)t≥0 in Wasserstein spaces follows the Kuwada’s duality approach [26], thanks to (1.3a), (1.3b) and the reﬁned argument developed in [5]. We can then prove the optim”
Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature (2016, background): “sharp for small |x−y| and/or large p. It is open whether (1.4), or in the Riemannian setting that RicZ is uniformly positive outside a compact domain, implies (1.3) for some constants c,λ>0. As in [15, 16], we will consider the Warsserstein distances induced by Young functions in the class N := n”
CALCULUS, HEAT FLOW AND CURVATURE-DIMENSION BOUNDS IN METRIC MEASURE SPACES (2019, methodology): “These inequalities can be improved, taking the dimension into account, in various ways, see Wang [ibid.] and the more recent papers Bolley, Gentil, and Guillin [2014] and Erbar, Kuwada, and Sturm [2015].”
Metric measure spaces with Riemannian Ricci curvature bounded from below (2011, background): “1) to conclude (see [26] for a generalization of this duality argument).”
Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case (2012, background): “By a duality argument (see [21], [15], [6]), this property implies the Bakry-Emery gradient estimate |Dht(f)|(2)w(x) ≤ eht(|Df |(2)w)(x), for m-a.”
Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope (2012, methodology): “This lemma has been introduced in [22] and then used in [12, 3] to study the heat flow on metric measure spaces.”
Bochner-Weitzenb\"ock formula and Li-Yau estimates on Finsler manifolds (2011, background): “Wasserstein space ([vRS, Corollary 1.4], see (1.1)). The equivalence between the gradient estimate and the contraction property is known to hold for more general linear semigroups by Kuwada’s duality [Ku]. Comparing Theorem 4.1 and Corollary 4.2 with the non-contraction property of heat ﬂow establis”
Heat Flow on Alexandrov Spaces (2010, background): “3), which follows with the aid of a result of the second author [19].”
The Splitting Theorem in Non-Smooth Context (2013, unknown): “Then one uses once again infinitesimal Hilbertianity to get that the flow ht, which coincides with the gradient flow of the energy f 7→ 1 2 ∫ |Df |2 dm, is linear and, by a general duality principle due to Kuwada (see [45]) to deduce that lip(ht(f)) 2 ≤ eht(lip(f)(2)), ∀f ∈ Lip(X) ∩ L(2)(X,m), then ”
Equivalent definitions of BV space and of total variation on metric measure spaces (2014, methodology): “This lemma, in the Wp case, has been introduced in [25] and then used in [17, 5] to study the heat flow on metric measure spaces.”