Naturally given any $s\in (0,1)$, the fractional Laplacian, $(-\Delta\_g)^s u$ on a closed Riemannian manifold can be defined via spectral decomposition of $-\Delta\_g$. There is another formulation of the fractional Laplacian that I commonly see that is stated as follows:
$$(-\Delta\_g)^sf(x)=\int\_0^{\infty} (e^{-t\Delta\_g}f(x)-f(x))\,t^{-1-s}\,dt.$$
Are these two definitions equivalent? Is there any reference for this?


Thanks,



Fractional Laplacian on closed manifolds
Yes, they are equivalent. Up to a constant missing and a sign error in the displayed formula, it should read:
$$ (-\Delta\_g)^s f(x) = \frac{1}{\Gamma(-s)} \int\_0^\infty (e^{t \Delta\_g} f(x) - f(x)) t^{-1-s} dt . $$
In fact, this is true for quite general operators. If $L^2$ theory is what you are after, this is a direct consequence of the spectral theorem and the gamma-type integral
$$\lambda^s = \frac{1}{\Gamma(-s)} \int\_0^\infty (e^{-\lambda t} - 1) t^{-1-s} dt . $$
If you are interested in more general spaces of functions, then *Bochner's subordination* is a right keyword. A standard reference is, I think, Martínez–Sans:


* C. Martínez, M. Sanz, *The Theory of Fractional Powers of Operators*.
North-Holland Math. Studies 187, Elsevier, Amsterdam (2001)


