  We analyze a quasi-continuous linear chain with self-similar distribution of
harmonic interparticle springs as recently introduced for one dimension
(Michelitsch et al., Phys. Rev. E 80, 011135 (2009)). We define a continuum
limit for one dimension and generalize it to $n=1,2,3,..$ dimensions of the
physical space. Application of Hamilton's (variational) principle defines then
a self-similar and as consequence non-local Laplacian operator for the
$n$-dimensional space where we proof its ellipticity and its accordance (up to
a strictly positive prefactor) with the fractional Laplacian
$-(-\Delta)^\frac{\alpha}{2}$. By employing this Laplacian we establish a
Fokker Planck diffusion equation: We show that this Laplacian generates
spatially isotropic L\'evi stable distributions which correspond to L\'evi
flights in $n$-dimensions. In the limit of large scaled times $\sim
t/r^{\alpha} >>1$ the obtained distributions exhibit an algebraic decay $\sim
t^{-\frac{n}{\alpha}} \rightarrow 0$ independent from the initial distribution
and spacepoint. This universal scaling depends only on the ratio $n/\alpha$ of
the dimension $n$ of the physical space and the L\'evi parameter $\alpha$.
