Equivalent semigroup properties for the curvature-dimension condition

By Feng-Yu Wang. 2010

55 citations recorded by the source corpus.

Some equivalent gradient and Harnack inequalities of a diffusion semigroup are presented for the curvature-dimension condition of the associated generator. As applications, the first eigenvalue, the log-Harnack inequality, the heat kernel estimates, and the HWI inequality are derived by using the curvature-dimension condition. The transportation inequality for diffusion semigroups is also investigated.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f8c9beca8bf1aca48101e8e45b4cfbd3ca7abd38

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality (2011, result): “Wang inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2660 3.3.”
Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds (2012, background): “2 proves a first L logL regularization estimate for the semigroup (Ht)t≥0, starting from arbitrary measures in P2(X) (here we follow the approach of [57]).”
On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, background): “See also the work of Wang [41], where the equivalence of gradient estimates and Bochner’s inequality has been rediscovered in the setting of smooth Riemannian manifolds.”
A PDE approach to a 2-dimensional matching problem (2016, background): “ be involved, in some statements we call a constant geometric if it depends only on Dthrough Csg, Cuc, Cge, Crt, Cdr and K. Notice that (GC) encodes a lower bound on Ricci curvature, see for instance [W11]. Let us draw now some easy consequences of these assumptions. Spectral gap implies that for f ”
Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces (2013, background): “ollows by choosing g:= fin (3.16). Corollary 3.5 Under the same assumption of Theorem 3.4, for every f∈ Vand α∈ [1/2,1] we have Γ Ptf α ≤ e−2αKtP t Γ f α . (3.23) Proof. We adapt here the strategy of [37]. Since the case α= 1 has been already covered by (3.2), we can also assume 1/2 ≤ α<1. We con”