Mediterr. J. Math. (2017) 14:192 DOI 10.1007/s00009-017-0992-9 c  Springer International Publishing AG 2017 Transportation Inequalities for Neutral Stochastic Differential Equations Driven by Fractional Brownian Motion with Hurst Parameter Lesser Than 1/2 Brahim Boufoussi and Salah Hajji Abstract. In this note, we prove an existence and uniqueness result of mild solution for a neutral stochastic differential equation with finite delay, driven by a fractional Brownian motion with Hurst parameter lesser than 1/2 in a Hilbert space, and we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution. Mathematics Subject Classification. 60H15, 60G22. 1. Introduction Let (E,d) be a metric space equipped with σ-field B, such that d(., .) is B×B- measurable. Given p ≥ 1 and two probability measures μ and ν on E, we define the Wasserstein distance of order p between μ and ν by W d p (μ, ν )= inf π∈Π(μ,ν)  E×E d(x, y) p dπ(x, y)  1/p , where Π(μ, ν ) is the set of all probability measures on the product space E × E with marginal distributions μ and ν . The relative entropy of ν with respect to μ is defined by the following: H(ν/μ)=  E log dν dμ dν, if ν ≪ μ +∞ otherwise. The probability measure μ satisfies the L p -transportation inequality on (E,d) if there exists a constant C ≥ 0, such that for any probability measure ν , W d p (μ, ν ) ≤  2CH(ν/μ). We shall write μ ∈ T p (C) for this relation. The cases “p = 1” and “p = 2” are of special interest. The phenomenon of measure concentration is related to T 1 (C). Based on the work of Bao et al. [2], Djellout et al.[4] proved that 0123456789().: V,-vol