  A proof is provided of a strong law of large numbers for a one-dimensional
random walk in a dynamic random environment given by a supercritical contact
process in equilibrium. The proof is based on a coupling argument that traces
the space-time cones containing the infection clusters generated by single
infections and uses that the random walk eventually gets trapped inside the
union of these cones. For the case where the local drifts of the random walk
are smaller than the speed at which infection clusters grow, the random walk
eventually gets trapped inside a single cone. This in turn leads to the
existence of regeneration times at which the random walk forgets its past. The
latter are used to prove a functional central limit theorem and a large
deviation principle. The qualitative dependence of the speed, the volatility
and the rate function on the infection parameter is investigated, and some open
problems are mentioned.
