Liggett and Steif (2006) proved that, for the supercritical contact process
\non certain graphs, the upper invariant measure stochastically dominates an
\ni.i.d.\\ Bernoulli product measure. In particular, they proved this for
\nmathbbZd and (for infection rate sufficiently large) d-ary homogeneous
\ntrees Td.
\n In this paper we prove some space-time versions of their results. We do this
\nby combining their methods with specific properties of the contact process and
\ngeneral correlation inequalities.
\n One of our main results concerns the contact process on Td with dgeq2.
\nWe show that, for large infection rate, there exists a subset Delta of the
\nvertices of Td, containing a "positive fraction" of all the vertices of
\nTd, such that the following holds: The contact process on Td observed on
\nDelta stochastically dominates an independent spin-flip process. (This is
\nknown to be false for the contact process on graphs having subexponential
\ngrowth.)
\n We further prove that the supercritical contact process on mathbbZd
\nobserved on certain d-dimensional space-time slabs stochastically dominates
\nan i.i.d.\\ Bernoulli product measure, from which we conclude strong mixing
\nproperties important in the study of certain random walks in random
\nenvironment