Recurrent Extensions of Real-Valued Self-Similar Markov Processes
- PDF / 481,365 Bytes
- 22 Pages / 439.642 x 666.49 pts Page_size
- 92 Downloads / 165 Views
Recurrent Extensions of Real-Valued Self-Similar Markov Processes H. Pant´ı1
· J. C. Pardo2 · V. M. Rivero2
Received: 31 July 2018 / Accepted: 26 June 2019 / © Springer Nature B.V. 2019
Abstract Let X = (Xt , t ≥ 0) be a self-similar Markov process taking values in R such that the state 0 is a trap. In this paper, we present a necessary and sufficient condition for the existence of a self-similar recurrent extension of X that leaves 0 continuously. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. Our results extend those of Fitzsimmons (Electron. Commun. Probab. 11, 230–241 2006) and Rivero (Bernoulli 11, 471–509 2005, 13, 1053–1070 2007) where the existence and uniqueness of a recurrent extension for positive self similar Markov processes were treated. In particular, we describe the recurrent extension of a stable L´evy process which to the best of our knowledge has not been studied before. Keywords Real self-similar Markov processes · Stable processes · Markov additive processes · Lamperti–Kiu representation · Exponential functional Mathematics Subject Classification (2010) 60G52 · 60G18 · 60G51
1 Introduction and main results In his seminal work [17], Lamperti studied the structure of positive self-similar Markov processes (pssMp) and posed the problem of determining those pssMp that agree with a given
H. Pant´ı
[email protected] J. C. Pardo [email protected] V. M. Rivero [email protected] 1
Facultad de Matem´aticas, Universidad Aut´onoma de Yucat´an, Anillo Perif´erico Norte, Tablaje Cat. 13615, Colonia Chuburn´a Hidalgo Inn, M´erida Yucat´an, M´exico
2
Centro de Investigaci´on en Matem´aticas, Calle Jalisco S/N. Col Valenciana, A.P. 402, C.P. 36000, Guanajuato, Guanajuato, M´exico
H. Pant´ı et al.
pssMp up to the time the latter process first hits 0. Lamperti [17] answered this question in the special case of Brownian motion killed at 0 and he found that the class of those extensions which are self-similar consists of the reflecting and absorbing Brownian motions and the extensions which immediately after reaching 0 jump according to the measure dx/x β+1 , β ∈ (0, 1). Voulle-Apiala [22] used Itˆo’s excursion theory to study the general case and provided a sufficient condition on the resolvent of pssMp for the existence of recurrent extensions that leave 0 continuously. The main contribution of Voulle-Apiala to this problem consist on the existence of a unique entrance law under which there exists a unique recurrent self-similar Markov process which turns out to be an extension of the pssMp after it reaches 0. Motivated by Voulle-Apiala’s result, Rivero [19] provided a simpler sufficient condition for the existence of such recurrent extension and a more explicit description of the entrance law. The sufficient condition found by Rivero was determined in terms of the underlying L´evy process in the so-called Lamperti’s transform of pssMp. Motivated by the aforementioned studies, Fitzsimmons [9] and Rivero [20] provided, independe
Data Loading...
