  We develop several results on hitting probabilities of random fields which
highlight the role of the dimension of the parameter space. This yields upper
and lower bounds in terms of Hausdorff measure and Bessel--Riesz capacity,
respectively. We apply these results to a system of stochastic wave equations
in spatial dimension $k\ge1$ driven by a $d$-dimensional spatially homogeneous
additive Gaussian noise that is white in time and colored in space.
