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The classical result due tof Williams states that a Brownian motion withpositive drift $\mu$ and issued from the origin is equal in law to a Brownianmotion with unit negative drift, $-\mu$, run until it hits a negativethreshold, whose depth below the origin is independently and exponentiallydistributed with parameter $2\mu$, after which it behaves like a Brownianmotion conditioned never to go below the aforesaid threshold (i.e. a Bessel-3process, or equivalently a Brownian motion conditioned to stay positive,relative to the threshold). In this article we consider the analogue ofWilliams' 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 describedas follows: suppose that we sample the point $x^*$ according to the law of thepoint of closest reach to the origin, sample; given $x^*$, we build$X^{\downarrow}$ having the law of $X$ conditioned to hit $x^*$ continuouslywithout entering the ball of radius $|x^*|$; then, we construct $X^\uparrow$ tohave the law of $X$ issued from $x^*$ conditioned never to enter the ball ofradius $|x^*|$; 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 originalssMp $X$. In essence, Williams' path decomposition in the setting of a ssMpfollows directly from an analogous decomposition for Markov additive processes(MAPs). The latter class are intimately related to the former via a space-timetransform known as the Lamperti--Kiu transform. As a key feature of our proofof Williams' path decomposition, will prove the analogue of Silverstein'sduality identity for the excursion occupation measure for general Markovadditive processes (MAPs).

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