Let X be a regular linear diffusion whose state space is an open interval
E⊆R. 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.Comment: 19 page