Parabolic Anderson model in bounded domains of recurrent metric measure spaces

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.