We derive stationary and fixation times for the multi-type Lambda Wright-Fisher process with and without mutations. Our method relies on a grand coupling of the process realized through the so-called lookdown-construction.

The classical result of 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 talk we will consider the analogue of Williams' path decomposition for a general self-similar Markov process (ssMp) on $\mathbb{R}^d$. 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 obtain the analogue of Silverstein's duality identity for the excursion occupation measure for general Markov additive processes (MAPs).