On Kelvin transformation

We prove that in the Euclidean space of arbitrary dimension the inversion of the isotropic stable Levy process killed at the origin is, after an appropriate change of time, the same stable process conditioned in the sense of Doob by the Riesz kernel. Using this identification we derive and explain transformation rules for the Kelvin transform acting on the Green function and the Poisson kernel of the stable process and on solutions of Schrodinger equation based on the fractional Laplacian. The Brownian motion and the classical Laplacian are included as a special case.