Abstract: Bañuelos and Bogdan (2004) and Bogdan, Palmowski and Wang (2016) analyse the asymptotic

tail distribution of the first time a stable (Lévy) process in dimension d\ge 2 exists a cone. We use these

results to develop the notion of a stable process conditioned to remain in a cone as well as the the notion

of a stable process conditioned to absorb continuously at the apex of a cone (without leaving the cone).

As self-similar Markov processes we examine some of their fundamental properties through the lens of

its Lamperti-Kiu decomposition. In particular we are interested to understand the underlying structure

of the Markov additive process that drives such processes. As a consequence of our interrogation of the

underlying MAP, we are able to provide an answer by example to the open question: If the modulator

of a MAP has a stationary distribution, under what conditions does its ascending ladder MAP have a

stationary distribution? We show how the two forms of conditioning are dual to one another.

Moreover, we construct the null-recurrent extension of the stable process killed on exiting a cone, showing that it again remains in the class of self-similar Markov processes.