A Free Analogue of the Transportation Cost Inequality on the Circle
نویسنده
چکیده
W ( ; ) := inf 2 ( ; ) sZZ 1 2 d(x; y)2 d (x; y); where d(x; y) = kx yk2 and ( ; ) denotes the set of all probability measures on R R with marginals and , i.e., ( R) = and (R ) = . The transportation cost inequality (TCI) obtained by M. Talagrand [15] is W ( ; ) p S( ; ) ; where is the standard Gaussian measure and is any probability measure on R. Recently Talagrand's inequality and its counterpart, the logarithmic Sobolev inequality have received a lot of attention and they have been extended from the Euclidean spaces to Riemannian manifolds. (Contrary to Talagrand's inequality, the logarithmic Sobolev inequality (LSI) gives an upper bound for the
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