2023/11/04 by Kyprianou, Andreas, Motala, Mehar, Rivero, Víctor · 1 citation
#60E10 #60J80 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2311.02375
The classical result due tof Williams states that a Brownian motion with positive drift μ and issued from the origin is equal in law to a Brownian motion with unit negative drift, -μ, run until it hits a negative threshold, whose depth below the origin is independently and exponentially distributed with parameter 2μ, 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 ℝd. Roughly speaking, we will prove that law of a ssMp, say X, in ℝd 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).