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,