STAPRO: 8203 Model 3G pp. 1–9 (col. fig: nil) Please cite this article in press as: Boufoussi B., Hajji S., Transportation inequalities for stochastic heat equations. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.03.012. Statistics and Probability Letters xx (xxxx) xxx–xxx Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Transportation inequalities for stochastic heat equations Brahim Boufoussi a , Salah Hajji b, * a Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 2390 Marrakesh, Morocco b Department of Mathematics, Regional Center for the Professions of Education and Training, Marrakesh, Morocco article info Article history: Received 18 November 2017 Received in revised form 6 March 2018 Accepted 16 March 2018 Available online xxxx MSC: 60H15 Keywords: Stochastic heat equation White noise Fractional noise Girsanov transformation Transportation inequality abstract Using the method of Girsanov transformation, we establish the transportation inequalities, on the continuous paths space with respect to the L 2 metric, for the law of the solution of stochastic heat equation driven by space–time white noise. The case of stochastic heat equations driven by fractional noise is also investigated. © 2018 Elsevier B.V. All rights reserved. 1. Introduction 1 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 2 measures μ and ν on E , we define the Wasserstein distance of order p between μ and ν by 3 W d p (μ,ν ) = inf π ∈Π(μ,ν) (∫ E×E d(x, y) p dπ (x, y) ) 1/p , 4 where Π (μ,ν ) is the set of all probability measures on the product space E × E with marginals μ and ν . The relative entropy 5 of ν with respect to μ is defined as 6 H(ν/μ) = ⎧ ⎨ ⎩ ∫ E log dν dμ dν, if ν ≪ μ +∞ otherwise. 7 The probability measure μ satisfies the L p − transportation inequality on (E , d) if there exists a constant C ≥ 0 such that 8 for any probability measure ν , 9 W d p (μ,ν ) ≤ √ 2CH(ν/μ). 10 We shall write μ ∈ T p (C ) for this relation. 11 * Corresponding author. E-mail addresses: boufoussi@ucam.ac.ma (B. Boufoussi), hajjisalahe@yahoo.fr (S. Hajji). https://doi.org/10.1016/j.spl.2018.03.012 0167-7152/© 2018 Elsevier B.V. All rights reserved.