No abstract

We show that any R d \{0}-valued self-similar Markov process X, with index α > 0 can be represented as a path transformation of some Markov additive process (MAP) (θ, ξ) in S d−1 × R. This result extends the well known Lamperti transformation. Let us denote by X the self-similar Markov process which is obtained from the MAP (θ, −ξ) through this extended Lamperti transformation. Then we prove that X is in weak duality with X, with respect to the measure π(x/ x ) x α−d dx, if and only if (θ, ξ) is reversible with respect to the measure π(ds)dx, where π(ds) is some σ-finite measure on S d−1 and dx is the Lebesgue measure on R. Besides, the dual process X has the same law as the inversionThese results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable Lévy processes.

No abstract

Let X be a regular linear diffusion whose state space is an open interval E ⊆ 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 3-dimensional hyperbolic Bessel process.