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  "body": "\n## Summary\n\nThe fractional Laplacian $(-\\Delta)^s$, for $0 \u003c s \u003c 1$, is a power of the operator $-\\Delta$ built from the [[Laplacian|0E0B6RA8K0]].\nIt acts on the [[Fourier transform|DPVM34ADZV]] side as multiplication by $\\abs{\\xi}^{2s}$.\nUnlike the Laplacian, it is nonlocal: $(-\\Delta)^s u(x)$ depends on the values of $u$ everywhere.\nIts negative $-(-\\Delta)^s$ is the generator of the rotationally symmetric $2s$-stable Lévy processes.\nIt appears in models of anomalous diffusion, in the thin obstacle problem and in [[fractional calculus|4V4AGE7XP4]].\n\n## Explanations\n\n### L1 human\n\nThe Laplacian compares the value of a function at a point with its average over tiny spheres around the point.\nThe fractional Laplacian compares it with values at all other points, with weights that decay like a power of the distance.\nSo a change far away still has a small effect here, which models particles that sometimes make long jumps.\nFor a wave $\\cos(k x)$ on the line, $-\\Delta$ multiplies it by $k^2$, and $(-\\Delta)^s$ multiplies it by $\\abs{k}^{2s}$.\nFor $s = \\tfrac12$ one gets a \"square root\" of $-\\Delta$, which on $\\R^n$ is the [[Dirichlet-to-Neumann operator|W783KDGT8E]] of the upper half-space $\\R^n \\times (0, \\infty)$.\n",
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