Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

Modified logarithmic Sobolev inequalities and transportation inequalities

“…Otto and Villani proved that a logarithmic Sobolev inequality implies a transportation inequality with a quadratic cost (this is the case α = β = 2), see [OV00,BGL01]. With the notations of this paper they prove that if µ satisfies the inequality LSI ·,2 (C), (when α = 2 the constant a is not any more a parameter in this case), then µ satisfies the inequality T ·,2 (4C).…”

“…When α = β = 2, we get as in [OV00], T ·,2 (C) → LSI ·,2 (16C). As in [OV00], Theorem 2.9 can be modified in the case Hess(ϕ) λId, where λ ∈ R.…”

Optimal transport and Perelman’s reduced volume

“…The rigorous proofs of the statements on P (M) are usually done by the Lagrangian approach, but one can also use the density of P ∞ (M) in P (M) [5,20]. Most of the calculations in this section can be extracted from [19] and [20].…”

“…In the associated Otto calculus, one considers P ∞ (M) to be an infinite-dimensional Riemannian manifold and E(c) to be the corresponding energy of the curve c, so the Euler-Lagrange equation for E becomes the geodesic equation on P ∞ (M) [18]. Otto and Villani used this approach to compute the time-derivatives of the entropy function E along the curve c [19].…”

Example of a displacement convex functional of first order

“…Carlen et al [4], have studied displacement convexity of the internal and the interaction energy in the setting of interfacial problems, where the configurations in the state space have infinite mass. Displacement convexity on Riemannian manifolds for internal energy (under Ricci curvature bounds) was predicted by Otto and Villani [21] and proved by Cordero-Erausquin, McCann, and Schmuckenschläger in [8]. Recently Otto and Westdickenberg [22] have introduced techniques that were further developed by Daneri and Savare [11] to show geodesic convexity of functionals on manifolds using a purely local, Eulerian framework.…”