Fractional Laplacian

Summary

The fractional Laplacian $(-\Delta)^s$, for $0 < s < 1$, is a power of the operator $-\Delta$ built from the Laplacian. It acts on the Fourier transform side as multiplication by $\abs{\xi}^{2s}$. Unlike the Laplacian, it is nonlocal: $(-\Delta)^s u(x)$ depends on the values of $u$ everywhere. Its negative $-(-\Delta)^s$ is the generator of the rotationally symmetric $2s$-stable Lévy processes. It appears in models of anomalous diffusion, in the thin obstacle problem and in fractional calculus.