Williams' path decomposition for self-similar Markov processes in $\mathbb{R}^d$

The classical result due tof Williams states that a Brownian motion with positive drift $\mu$ and issued from the origin is equal in law to a Brownian motion with unit negative drift, $-\mu$, run until it hits a negative threshold, whose depth below the origin is independently and exponentially distributed with parameter $2\mu$, 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 $\mathbb{R}^d$. Roughly speaking, we will prove that law of a ssMp, say $X$, in $\mathbb{R}^d$ 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, sample; given $x^*$, we build $X^{\downarrow}$ having the law of $X$ conditioned to hit $x^*$ continuously without entering the ball of radius $|x^*|$; then, we construct $X^\uparrow$ to have the law of $X$ issued from $x^*$ conditioned never to enter the ball of radius $|x^*|$; glueing the path of $X^\uparrow$ end-to-end with $X^\downarrow$ 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 for general Markov additive processes (MAPs).