  We study concentration properties for laws of non-linear Gaussian functionals
on metric spaces. Our focus lies on measures with non-Gaussian tail behaviour
which are beyond the reach of Talagrand's classical Transportation-Cost
Inequalities (TCIs). Motivated by solutions of Rough Differential Equations and
relying on a suitable contraction principle, we prove generalised TCIs for
functionals that arise in the theory of regularity structures and, in
particular, in the cases of rough volatility and the two-dimensional Parabolic
Anderson Model. In doing so, we also extend existing results on TCIs for
diffusions driven by Gaussian processes.
