  We develop criteria for hitting probabilities of anisotropic Gaussian random
fields with associated canonical pseudo-metric given by a class of gauge
functions. This yields lower and upper bounds in terms of general notions of
capacity and Hausdorff measure, respectively, therefore extending the classical
estimates with the Bessel-Riesz capacity and the $\gamma$-dimensional Hausdorff
measure. We apply the criteria to a system of linear stochastic partial
differential equations driven by space-time noises that are fractional in time
and either white or colored in space.

Probability