Transportation cost for Gaussian and other product measures. (English) Zbl 0859.46030

Talagrand, M.

Geom. Funct. Anal. 6, No. 3, 587-600 (1996).

Summary: Consider the canonical Gaussian measure \(\gamma_N\) on \(\mathbb{R}^N\), a probability measure \(\mu\) on \(\mathbb{R}^N\), absolutely continuous with respect to \(\gamma_N\). We prove that the transportation cost of \(\mu\) to \(\gamma_N\), when the cost of transporting a unit of mass from \(x\) to \(y\) is measured by \(|x-y|^2\), is at most \(\int\log{d\mu\over d\gamma_N} d\mu\). As a consequence we obtain a completely elementary proof of a very sharp form of the concentration of measure phenomenon in Gauß space. We then prove a result of the same nature when \(\gamma_N\) is replaced by the measure of density \(2^{-N}\exp(-\sum_{i\leq N}|x_i|)\). This yields a sharp form of concentration of measure in that space.

MSC:

46G12	Measures and integration on abstract linear spaces
60B11	Probability theory on linear topological spaces
28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

Keywords:

canonical Gaussian measure; probability measure; transportation cost; concentration of measure phenomenon in Gauß space

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