  A metric measure space equipped with a Dirichlet form is called recurrent if
its Hausdorff dimension is less than its walk dimension. In bounded domains of
such spaces we study the parabolic Anderson models \[ \partial_{t} u(t,x) =
\Delta u(t,x) + \beta u(t,x) \, \dot{W}_\alpha(t,x) \] where the noise
$W_\alpha$ is white in time and colored in space when $\alpha >0$ while for
$\alpha=0$ it is also white in space. Both Dirichlet and Neumann boundary
conditions are considered. Besides proving existence and uniqueness in the
It\^o sense we also get precise $L^p$ estimates for the moments and
intermittency properties of the solution as a consequence. Our study reveals
new exponents which are intrinsically associated to the geometry of the
underlying space and the results for instance apply in metric graphs or
fractals like the Sierpi\'nski gasket for which we prove scaling invariance
properties of the models.
