  We study the spectral dimension associated with diffusion processes on
Euclidean $\kappa$-Minkowski space. We start by describing a geometric
construction of the "Euclidean" momentum group manifold related to
$\kappa$-Minkowski space. On such space we identify various candidate Laplacian
functions, i.e. deformed Casimir invariants, and calculate the corresponding
spectral dimension for each case. The results obtained show a variety of
running behaviours for the spectral dimension according to the choice of
deformed Laplacian, from dimensional reduction to super-diffusion.
