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Let $X$ be a standard Markov process. We prove that a space inversionproperty of $X$ implies the existence of a Kelvin transform of $X$-harmonic,excessive and operator-harmonic functions and that the inversion property isinherited by Doob $h$-transforms. We determine new classes of processes havingspace inversion properties amongst transient processes {satisfying the} timeinversion property. {For these processes, some explicit inversions, which areoften not the spherical ones, and excessive functions are given explicitly.} Wetreat in details the examples of free scaled power Bessel processes,non-colliding Bessel particles, Wishart processes, Gaussian Ensemble and DysonBrownian Motion.

Space and time inversions of stochastic processes and Kelvin transform