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Abstract

Let ( X , ρ ) be a Polish space endowed with a probability measure μ . Assume that we can do Malliavin Calculus on ( X , μ ) . Let d : X × X → [ 0 , + ∞ ] be a pseudo-distance. Consider Q t F ( x ) = inf y ∈ X { F ( y ) + d 2 ( x , y ) / 2 t } . We shall prove that Q t F satisfies the Hamilton–Jacobi inequality under suitable conditions. This result will be applied to establish transportation cost inequalities on path groups and loop groups in the spirit of Bobkov, Gentil and Ledoux.