Concerning the semigroup connection, the User’s guide to the fractional Laplacian and the method of semigroups (2018) may provide a helpful entry point to the literature:
The fractional Laplacian $L^s=(-\Delta)^s$, $0<s<1$ can be expressed in terms of the heat diffusion semigroup $v=e^{-tL}u$ generated by $L$ acting on $u$ through the integral formula $$L^s u=\frac{1}{\Gamma(-s)}\int_0^\infty\left(e^{-tL}u-u\right)\frac{dt}{t^{1+s}}.$$ The solution to $L^s u=f$ can then be written as $$u=\frac{1}{\Gamma(s)}\int_0^\infty e^{-tL}f\frac{dt}{t^{1-s}}.$$