  We consider a random walk on top of the contact process on $\mathbb{Z}^d$
with $d\geq 1$. In particular, we focus on the "contact process as seen from
the random walk". Under the assumption that the infection rate of the contact
process is large or the jump rate of the random walk is small, we show that
this process has at most two extremal measures. Moreover, the convergence to
these extremal measures is characterised by whether the contact process
survives or dies out, similar to the complete convergence theorem known for the
ordinary contact process. Using this, we furthermore provide a law of large
numbers for the random walk which holds under general assumptions on the jump
probabilities of the random walk.

Probability