Talagrand's transportation-cost inequality and applications to (rough) path spaces

We give a new proof of Talagrand's transportation-cost inequality on Gaussian spaces. The proof combines the large deviation approach from Gozlan in [Ann. Prob. (2009)] with the Borell-Sudakov-Tsirelson inequality. Several applications are discussed. First, we show how to deduce transportation-cost inequalities for the law of diffusions driven by Gaussian processes both in the additive and the multiplicative noise case. In the multiplicative case, the equation is understood in rough paths sense and we use properties of the It\=o-Lyons map to deduce the inequalities which improves existing results even in the Brownian motion case. Second, we present a general theorem which allows to derive Gaussian tail estimates for functionals on spaces on which a $p$-transportation-cost inequality holds. In the Gaussian case, this result can be seen as a generalization of the ``generalized Fernique theorem'' from Friz and Oberhauser obtained in [Proc. Amer. Math. Soc. (2010)]. Applications to objects in rough path theory are given, such as solutions to rough differential equations and to a counting process studied by Cass, Litterer, Lyons in [Ann. Prob. (2013)].