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,
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:
