Larbi Alili: On some involutive inversions of one dimensional transient diffusions

Given a regular one dimensional diffusion on an interval, where regular boundaries are killing, we show that, in some sense, this has a unique inverse with respect to a fixed point in the state space. This inverse process when appropriately time-changed is the dual process which law is obtained by Doob h-transforming the law of the original diffusion. Conversely, given a positive harmonic function h on that interval (or a scale function), satisfying some reasonable conditions, we show that the Doob h-transformed process can be obtained explicitly in terms of inversion of the original diffusion with a time change. The construction involves some interesting involutions and random clocks.

This is joint work with P. Graczyk and T. Zak.