On Inversions and Doob H-Transforms of Linear Diffusions

Let X be a regular linear diffusion whose state space is an open interval E subset of R. We consider the dual diffusion X* whose 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. We provide a construction of X* as a deterministic inversion I(X) of X, time changed with some random clock. Such inversions generalize the Euclidean inversions that intervene when X is a Brownian motion. The important case where X* is X conditioned to stay above some fixed level is included. The families of deterministic inversions are given explicitly for the Brownian motion with drift, Bessel processes and the three-dimensional hyperbolic Bessel process.