Inversion, duality and Doob $h$-transforms for self-similar Markov processes

Larbi Alili,Tomasz Żak,Loïc Chaumont,Piotr Graczyk

doi:10.1214/17-ejp33

Abstract

We show that any Rd∖{0}Rd∖{0}-valued self-similar Markov process XX, with index α>0α>0 can be represented as a path transformation of some Markov additive process (MAP) (θ,ξ)(θ,ξ) in Sd−1×RSd−1×R. This result extends the well known Lamperti transformation. Let us denote by XˆX^ the self-similar Markov process which is obtained from the MAP (θ,−ξ)(θ,−ξ) through this extended Lamperti transformation. Then we prove that XˆX^ is in weak duality with XX, with respect to the measure π(x/∥x∥)∥x∥α−ddxπ(x/‖x‖)‖x‖α−ddx, if and only if (θ,ξ)(θ,ξ) is reversible with respect to the measure π(ds)dxπ(ds)dx, where π(ds)π(ds) is some σσ-finite measure on Sd−1Sd−1 and dxdx is the Lebesgue measure on RR. Moreover, the dual process XˆX^ has the same law as the inversion (Xγt/∥Xγt∥2,t≥0)(Xγt/‖Xγt‖2,t≥0) of XX, where γtγt is the inverse of t↦∫t0∥X∥−2αsdst↦∫0t‖X‖s−2αds. These results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable Levy processes.

Highlights

There exist many ways to construct the three dimensional Bessel process from Brownian motion
We show that a necessary and sufficient condition for this to hold is that the underlying Markov additive process in the Lamperti representation satisfies a condition of reversibility
[20], who proved that all positive self-similar Markov processes can be obtained as exponentials of time changed Levy process

Summary

Introduction

There exist many ways to construct the three dimensional Bessel process from Brownian motion. Our result implies Theorem 4.1 of Lamperti [20], who proved that all positive self-similar Markov processes can be obtained as exponentials of time changed Levy process. In this case, in order to describe the behaviour of {X, Px} at its lifetime, it suffices to note from general properties of. Let {X, Px} be an Rd-valued Markov process satisfying the scaling property (2.2) with α > 0, and assume that it has an infinite lifetime.This means in particular that {X, Px} can possibly hit 0 without being absorbed, like real Brownian motion for instance.

Inversion and duality of ssMp’s

Inversion and Doob h-transforms for ssMp’s

Conditioned Levy processes

Free d-dimensional Bessel processes

Dunkl processes