  We investigate small deviation properties of Gaussian random fields in the
space $L_q(\R^N,\mu)$ where $\mu$ is an arbitrary finite compactly supported
Borel measure. Of special interest are hereby "thin" measures $\mu$, i.e.,
those which are singular with respect to the $N$--dimensional Lebesgue measure;
the so--called self--similar measures providing a class of typical examples.
  For a large class of random fields (including, among others, fractional
Brownian motions), we describe the behavior of small deviation probabilities
via numerical characteristics of $\mu$, called mixed entropy, characterizing
size and regularity of $\mu$.
  For the particularly interesting case of self--similar measures $\mu$, the
asymptotic behavior of the mixed entropy is evaluated explicitly. As a
consequence, we get the asymptotic of the small deviation for $N$--parameter
fractional Brownian motions with respect to $L_q(\R^N,\mu)$--norms.
  While the upper estimates for the small deviation probabilities are proved by
purely probabilistic methods, the lower bounds are established by analytic
tools concerning Kolmogorov and entropy numbers of H\"older operators.
