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We analyze a quasi-continuous linear chain with self-similar distribution ofharmonic interparticle springs as recently introduced for one dimension(Michelitsch et al., Phys. Rev. E 80, 011135 (2009)). We define a continuumlimit for one dimension and generalize it to $n=1,2,3,..$ dimensions of thephysical space. Application of Hamilton's (variational) principle defines thena self-similar and as consequence non-local Laplacian operator for the$n$-dimensional space where we proof its ellipticity and its accordance (up toa strictly positive prefactor) with the fractional Laplacian$-(-\Delta)^\frac{\alpha}{2}$. By employing this Laplacian we establish aFokker Planck diffusion equation: We show that this Laplacian generatesspatially isotropic L\'evi stable distributions which correspond to L\'eviflights in $n$-dimensions. In the limit of large scaled times $\simt/r^{\alpha} >>1$ the obtained distributions exhibit an algebraic decay $\simt^{-\frac{n}{\alpha}} \rightarrow 0$ independent from the initial distributionand spacepoint. This universal scaling depends only on the ratio $n/\alpha$ ofthe dimension $n$ of the physical space and the L\'evi parameter $\alpha$.

An approach to anomalous diffusion in the n-dimensional space generated by a self-similar Laplacian