Conditioned real self-similar Markov processes

Stable Lévy processes, self-similarity and the unit ball

“…Just as in Kyprianou et al (2015) it can be shown that P • x , x ∈ R d \{0}, again corresponds to the law of conditioning the stable process to continuously absorb a the origin. The origins of Proposition 3.14 can already be found in Bogdan and Żak (2006).…”

“…Stable processes killed on entering (−∞, 0). This first example was introduced in detail in Caballero and Chaumont (2006a); see also Kyprianou et al (2015). To some extent, the former of these two references marks the beginning of the modern treatment of stable processes through the theory of self-similar Markov processes.…”

“…This comes about by a change of measure, which corresponds to a Doob h-transform to the semigroup of a two-sided jumping stable process killed on first hitting the origin if α ∈ (1, 2). As it is instructive for future discussion, we give a proof of the following result which is originally from Chaumont et al (2013), for α ∈ (1, 2), and Kyprianou et al (2015), for α ∈ (1, 2). Our proof differs slightly from its original setting.…”

“…It penalises those paths that approach the origin and rewards those that stray away from the origin. In fact, it has been shown in Kyprianou et al (2015) that, for α ∈ (0, 1), in the appropriate sense, the change of measure is equivalent to conditioning the stable process to continuously absorb at the origin, and when α ∈ (1, 2), in Chaumont et al (2013); Kuznetsov et al (2014) it is shown that the change of measure is equivalent to conditioning the stable process to avoid the origin.…”

“…With only the sporadic works of e.g. Graversen and Vuolle-Apiala (1986); Vuolle-Apiala and ; Kiu (1980) in the 1980s, only very recently has the bigger picture begun to emerge some 30 or more years later in Xiao (1998); Nane et al (2010); Chaumont et al (2013); Kuznetsov et al (2014); Alili et al (2017); Kyprianou et al (2015); Kyprianou (2016); ).…”