Williams’ path decomposition for self-similar Markov processes in R^d

The classical result of Williams [27] states that a Brownian motion with positive drift µ and issued from the origin is equal in law to a Brownian motion with unit negative drift, −µ, run until it hits a negative threshold, whose depth below the origin is independently and exponentially distributed with parameter 2µ, after which it behaves like a Brownian motion conditioned never to go below the aforesaid threshold (i.e. a Bessel-3 process, or equivalently a Brownian motion conditioned to stay positive, relative to the threshold). In this article we consider the analogue of Williams’ path decomposition for a general self-similar Markov process (ssMp) on R^d. Roughly speaking, we will prove that the law of a ssMp, say X, in R^d is equivalent in law to the concatenation of paths described as follows: suppose that we sample the point x^∗ according to the law of the point of closest reach to the origin; given x^∗, we build X^↓ having the law of X conditioned to hit x^∗ continuously without entering the ball of radius |x^∗ |; then, we construct X^↑ to have the law of X issued from x^∗ conditioned never to enter the ball of radius |x^∗ |; glueing the path of X^↑ end-to-end with X^↓ via the point x^∗ produces a process which is equal in law to our original ssMp X. In essence, Williams’ path decomposition in the setting of a ssMp follows directly from an analogous decomposition for Markov additive processes (MAPs). The latter class are intimately related to the former via a space-time transform known as the Lamperti–Kiu transform. As a key feature of our proof of Williams’ path decomposition, will prove the analogue of Silverstein’s duality identity for the excursion occupation measure, cf. [26], for general Markov additive processes (MAPs).