The Annals of Probability

Real self-similar processes started from the origin

Steffen Dereich, Leif Döring, and Andreas E. Kyprianou

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Since the seminal work of Lamperti, there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than $0$ which subsequently was extended to zero initial condition.

For real self-similar Markov processes (rssMps), there is a generalization of Lamperti’s representation giving a one-to-one correspondence between Markov additive processes and rssMps with initial condition different from the origin.

We develop fluctuation theory for Markov additive processes and use Kuznetsov measures to construct the law of transient real self-similar Markov processes issued from the origin. The construction gives a pathwise representation through two-sided Markov additive processes extending the Lamperti–Kiu representation to the origin.

Article information

Ann. Probab., Volume 45, Number 3 (2017), 1952-2003.

Received: December 2014
First available in Project Euclid: 15 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G18: Self-similar processes 60G51: Processes with independent increments; Lévy processes
Secondary: 60B10: Convergence of probability measures 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Self-similar process Markov additive process fluctuation theory


Dereich, Steffen; Döring, Leif; Kyprianou, Andreas E. Real self-similar processes started from the origin. Ann. Probab. 45 (2017), no. 3, 1952--2003. doi:10.1214/16-AOP1105.

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  • [1] Alsmeyer, G. (1994). On the Markov renewal theorem. Stochastic Process. Appl. 50 37–56.
  • [2] Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Springer, New York.
  • [3] Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press, Cambridge.
  • [4] Bertoin, J. and Savov, M. (2011). Some applications of duality for Lévy processes in a half-line. Bull. Lond. Math. Soc. 43 97–110.
  • [5] Chaumont, L. and Doney, R. A. (2005). On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 948–961.
  • [6] Chaumont, L., Kyprianou, A., Pardo, J. C. and Rivero, V. (2012). Fluctuation theory and exit systems for positive self-similar Markov processes. Ann. Probab. 40 245–279.
  • [7] Chaumont, L., Pantí, H. and Rivero, V. (2013). The Lamperti representation of real-valued self-similar Markov processes. Bernoulli 19 2494–2523.
  • [8] Dellacherie, C., Maisonneuve, B. and Meyer, P. A. (1992). Probabilités et Potentiel: Processus de Markov (Fin). Compléments du Calcul Stochastique. Hermann, Paris.
  • [9] Doney, R. A. and Maller, R. A. (2002). Stability and attraction to normality for Lévy processes at zero and at infinity. J. Theoret. Probab. 15 751–792.
  • [10] Döring, L. (2015). A jump-type SDE approach to real-valued self-similar Markov processes. Trans. Amer. Math. Soc. 367 7797–7836.
  • [11] Fitzsimmons, P. J. (1988). On a connection between Kuznetsov processes and quasi-processes. In Seminar on Stochastic Processes, 1987 (Princeton, NJ, 1987). Progr. Probab. Statist. 15 123–133. Birkhäuser, Boston, MA.
  • [12] Fitzsimmons, P. J. (2006). On the existence of recurrent extensions of self-similar Markov processes. Electron. Commun. Probab. 11 230–241.
  • [13] Fitzsimmons, P. J. and Maisonneuve, B. (1986). Excessive measures and Markov processes with random birth and death. Probab. Theory Related Fields 72 319–336.
  • [14] Fitzsimmons, P. J. and Maisonneuve, B. (1986). Excessive measures and Markov processes with random birth and death. Probab. Theory Related Fields 72 319–336.
  • [15] Fitzsimmons, P. J. and Taksar, M. (1988). Stationary regenerative sets and subordinators. Ann. Probab. 16 1299–1305.
  • [16] Jacobsen, M. and Yor, M. (2003). Multi-self-similar Markov processes on ${\mathbb{R}}^{n}_{+}$ and their Lamperti representations. Probab. Theory Related Fields 126 1–28.
  • [17] Kaspi, H. (1982). On the symmetric Wiener–Hopf factorization for Markov additive processes. Z. Wahrsch. Verw. Gebiete 59 179–196.
  • [18] Kaspi, H. (1988). Random time changes for processes with random birth and death. Ann. Probab. 16 586–599.
  • [19] Kesten, H. (1974). Renewal theory for functionals of a Markov chain with general state space. Ann. Probab. 2 355–386.
  • [20] Kiu, S. W. (1980). Semistable Markov processes in $\textbf{R}^{n}$. Stochastic Process. Appl. 10 183–191.
  • [21] Kuznetsov, A., Kyprianou, A. E., Pardo, J. C. and Watson, A. R. (2014). The hitting time of zero for a stable process. Electron. J. Probab. 19 no. 30, 26.
  • [22] Kuznetsov, S. E. (1974). In construction of Markov processes with random birth and death. Theory Probab. Appl. 18 571–575.
  • [23] Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2nd ed. Springer, Heidelberg.
  • [24] Lalley, S. P. (1984). Conditional Markov renewal theory. I. Finite and denumerable state space. Ann. Probab. 12 1113–1148.
  • [25] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 205–225.
  • [26] Mitro, J. B. (1979). Dual Markov processes: Construction of a useful auxiliary process. Z. Wahrsch. Verw. Gebiete 47 139–156.
  • [27] Rivero, V. (2007). Recurrent extensions of self-similar Markov processes and Cramér’s condition. II. Bernoulli 13 1053–1070.
  • [28] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge.