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A new characterization of Talagrand’s transport-entropy inequalities and applications. (English) Zbl 1233.60007

The starting point of the paper under review is Talagrand’s transportation cost inequality for \(\mu, \nu\in M^1(\mathbb{R}^k)\), a cost function \(c:\mathbb{R}^k\to\mathbb{R}_+\) (with \(c(0)=\nabla c(0)=0\)) and a constant \(C>0\): \[ T_c(\nu,\mu) \leq C\cdot H(\nu|\mu), \tag{\(T_c(C)\)} \] where the optimal transport costs are defined as
\[ T_c(\nu,\mu) = \inf_\pi \iint c(x-y)d\pi(x,y), \]
the infimum is taken over all \(\pi\in M^1(\mathbb{R}^{2k})\) with marginals \(\nu\) and \(\mu\), respectively, and the relative entropy is respectively defined as \(H(\nu|\mu)=\int \log(d\nu/d\mu) d\nu\) if \(\nu\) is absolutely continuous with respect to \(\mu\), and \(H(\nu|\mu)=\infty\) otherwise, cf., e.g., for Gaussian laws and quadratic cost functions [M. Talagrand, “Transportation cost for Gaussian and other product measures”, Geom. Funct. Anal. 6, No. 3, 587–600 (1996; Zbl 0859.46030)]. These inequalities are closely related to logarithmic Sobolev inequalities, c.f., e.g., [F. Otto and C. Villani, “Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality”, J. Funct. Anal. 173, No. 2, 361–400 (2000; Zbl 0985.58019)].
In the present investigation, the authors prove (among other results of independent interest) the equivalence of \((T_c(C_1))\) with restricted versions of logarithmic Sobolev inequalities (Theorem 1.5), as \[ \text{Ent}_\mu(e^f) \leq \eta/(1-C_2(\eta+K))\int c^*(\nabla f/\eta)e^f d\mu \] and \[ \text{Ent}_\mu(e^f) \leq 2C_3/(1-KC_3)^2 \int |\nabla f|^2 e^f d\mu \] for \(f\) belonging to a class of functions (\(K\) semi convex functions) \(K, C_i,\eta\) denoting suitable positive constants and
\[ \begin{aligned}\text{Ent}_\mu(g)&:=\int g\log g d\mu - \int g d\mu \log\int g d\mu, \\ c^*(u)&:= \sup_{h\in\mathbb{R}^k} \{\langle u,h\rangle -c(h)\}, \end{aligned} \]
and particular cost functions \(c(x):=\sum_1^k\alpha_i(x_i)\) with concave \(\alpha_i\in C^1(\mathbb{R})\), \(\alpha_i(0)=\alpha_i'(0)=0\).
The paper closes with possible extensions of the main results, as extensions to Riemannian manifolds, relations to other types of logarithmic Sobolev inequalities, and to Poincaré’s inequality. (For background, see, e.g., [S. G. Bobkov, I. Gentil and M. Ledoux, “Hypercontractivity of Hamilton-Jacobi equations”, J. Math. Pures Appl. (9) 80, No. 7, 669–696 (2001; Zbl 1038.35020); S. G. Bobkov and M. Ledoux, “Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution”, Probab. Theory Relat. Fields 107, No. 3, 383–400 (1997; Zbl 0878.60014)]).

MSC:

60E15 Inequalities; stochastic orderings
60F10 Large deviations
26D10 Inequalities involving derivatives and differential and integral operators
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