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 QtF(x) = infy∈X{F(y) + d2(x, y)/2t}. We shall prove that QtF 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.