We continue our investigation on the transportation-information inequalities
WpI for a symmetric markov process, introduced and studied in \cite{GLWY}.
We prove that WpI implies the usual transportation inequalities WpH, then
the corresponding concentration inequalities for the invariant measure μ.
We give also a direct proof that the spectral gap in the space of Lipschitz
functions for a diffusion process implies W1I (a result due to \cite{GLWY})
and a Cheeger type's isoperimetric inequality. Finally we exhibit relations
between transportation-information inequalities and a family of functional
inequalities (such as Φ-log Sobolev or Φ-Sobolev)