  Inequalities between transportation costs and Fisher information are known to
characterize certain concentration properties of Markov processes around their
invariant measures. This note provides a new characterization of the quadratic
transportation-information inequality $W_2I$ in terms of a dimension-free
concentration property for i.i.d. (conditionally on the initial positions)
copies of the underlying Markov process. This parallels Gozlan's
characterization of the quadratic transportation-entropy inequality $W_2H$. The
proof is based on a new Laplace-type principle for the operator norms of
Feynman-Kac semigroups, which is of independent interest. Lastly, we illustrate
how both our theorem and (a form of) Gozlan's are instances of a general
convex-analytic tensorization principle.
