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

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.


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.)
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