Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces

By Giuseppe Savar'e. Discrete & Continuous Dynamical Systems - A, 2013

213 citations recorded by the source corpus.

We prove that the linear ``heat'' flow in a $RCD (K, \infty)$ metric measure space $(X, d, m)$ satisfies a contraction property with respect to every $L^p$-Kantorovich-Rubinstein-Wasserstein distance, $p\in [1,\infty]$. In particular, we obtain a precise estimate for the optimal $W_\infty$-coupling between two fundamental solutions in terms of the distance of the initial points. The result is a consequence of the equivalence between the $RCD (K, \infty)$ lower Ricci bound and the corresponding Bakry-Emery condition for the canonical Cheeger-Dirichlet form in $(X, d, m)$. The crucial tool is the extension to the non-smooth metric measure setting of the Bakry's argument, that allows to improve the commutation estimates between the Markov semigroup and the Carre du Champ $\Gamma$ associated to the Dirichlet form. This extension is based on a new a priori estimate and a capacitary argument for regular and tight Dirichlet forms that are of independent interest.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d57256560ec24418d9f4b5dcc84ff2f3e50f6495

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Citations in the corpus

Obata’s Rigidity Theorem for Metric Measure Spaces (2014, background): “4 in [24] to obtain a self-improved, sharp Γ2-estimate that also involves terms depending on the dimension.”
Structure theory of metric measure spaces with lower Ricci curvature bounds (2014, background): “While a great deal of progress has been made in this latter general framework, see for instance [3, 5, 6, 7, 9, 10, 11, 15, 17, 29, 30, 31, 32, 34, 35, 36, 42, 43, 44, 48], the structure theory on such metric-measure spaces is still much less developed than in the case of smooth limits.”
The Splitting Theorem in Non-Smooth Context (2013, unknown): “It is unclear to us if this can really be done, part of the problem being to define what the Hessian is, but we point out that Honda in [41] proved that some sort of second order differential structure exists on spaces which are limits of Riemannian manifolds with Ricci curvature uniformly bounded f”
Second order differentiation formula on RCD*$(K,N)$ spaces (2018, background): “We remark that in [37] the authors refer to [27], [3], [6] and [51] for the various calculus rules and that in these latter references no properness assumption is made.”
Independence on p of weak upper gradients on RCD spaces (2014, background): “The argument is based on some regularization properties of the heat flow proved in [9] and on the density in energy of Lipschitz functions in Sobolev spaces established in [3].”
On Nilpotent Subsemigroups in some Matrix Semigroups (2005, unknown): “Hence, taking account of the existing literature, estimate (3) represents a generalisation of all the above results and the purely analytical proof we proposed, inspired by an idea due to Bakry and ´ Emery (see [4] and [38]), is a novelty in the proofs of gradient estimates.”
On the Bakry-\'Emery condition, the gradient estimates and the Local-to-Global property of RCD*(K,N) metric measure spaces (2013, unknown): “Improving some results of [31], in this paper we show that in BE(K ,∞) spaces, functions f ∈ L2 ∩ L∞(X,m) with f ∈ L2(X,m) satisfy the extra integrability property for ( f )”
From volume cone to metric cone in the nonsmooth setting (2015, methodology): “On a RCD∗(0, N) space we consider the following space of test functions (see [23] and [13]): Test(X) := { f ∈ D(∆) : f, |Df | ∈ L∞(X), ∆f ∈W (1,2)(X) } .”
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces (2015, result): “[16, 7, 46], and the mass preserving property (3.”