  We start by considering infinite dimensional Markovian dynamics in R^m
generated by operators of hypocoercive type and for such models we obtain short
and long time pointwise estimates for all the derivatives, of any order and in
any direction, along the semigroup. We then look at infinite dimensional models
(in (Rm)^{Z ^d}) produced by the interaction of infinitely many finite
dimensional dissipative dynamics of the type indicated above. For these
infinite dimensional models we study finite speed of propagation of
information, well-posedness of the semigroup, time behaviour of the derivatives
and strong ergodicity problem.
