On inversions and Doob h-transforms of linear diffusions

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Let X be a regular linear diffusion whose state space is an open interval E. We consider a diffusion X* which probability law is obtained as a Doob h-transform of the law of X, where h is a positive harmonic function for the infinitesimal generator of X on E. This is the dual of X with respect to h(x)m(dx) where m(dx) is the speed measure of X. Examples include the case where X* is X conditioned to stay above some fixed level. We provide a construction of X* as a deterministic inversion of X, time changed with some random clock. The study involves the construction of some inversions which generalize the Euclidean inversions. Brownian motion with drift and Bessel processes are considered in details.