zbMATH — the first resource for mathematics

Infinitely ramified point measures and branching Lévy processes. (English) Zbl 1456.60225
In analogy to the well-known relation between infinitely divisible distributions and processes with stationary independent increments (Lévy processes), the authors connect what they call infinitely ramified point measures (IRPM) with branching Lévy processes (BLP). An IRPM is defined as a random point measure \(\mathcal{Z}\) which for every \(n\in\mathbb{N}\) has the same distribution as the \(n\)th generation of some branching random walk. In the considered BLP, particles move independently according to Lévy process and produce progeny during their lifetime similarly as in a Crump-Mode-Jagers branching process. The point measures, random walks and Lévy processes are taken here on the real line. Denote \(\langle\Sigma_n \delta_{x_n},f\rangle:= \Sigma_n f(x_n)\) and \(\textbf{e}_\theta(x) :=\textbf{e}^{x\theta}\), \(\theta\ge 0\), \(x\in\mathbb{R}\). It is shown that given an IRPM \(\mathcal{Z}\) such that (*) \(0< E(\langle\mathcal{Z}, \textbf{e}_\theta\rangle)<\infty\) for some \(\theta\ge 0\), there exists a BLP \(Z= \{Z_t; t\ge 0\}\) with \(\mathcal{Z}\overset{(d)}{=} Z_1\). Vice versa, if \(Z\) is a BLP such that the corresponding Lévy measure satisfies certain integrability conditions, then \(Z_1\) is an IRPM satisfying (*).

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G51 Processes with independent increments; Lévy processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
Full Text: DOI Euclid arXiv
[1] Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY. · Zbl 1070.60001
[2] Bertoin, J. (2016). Compensated fragmentation processes and limits of dilated fragmentations. Ann. Probab.44 1254–1284. · Zbl 1344.60033
[3] Bertoin, J. and Rouault, A. (2005). Discretization methods for homogeneous fragmentations. J. Lond. Math. Soc. (2) 72 91–109. · Zbl 1077.60053
[4] Bovier, A. (2017). Gaussian Processes on Trees: From Spin Glasses to Branching Brownian Motion. Cambridge Studies in Advanced Mathematics163. Cambridge Univ. Press, Cambridge. · Zbl 1378.60004
[5] Brooks, J. K. and Dinculeanu, N. (1987). Projections and regularity of abstract processes. Stoch. Anal. Appl.5 17–25. · Zbl 0619.60038
[6] Chauvin, B. (1991). Product martingales and stopping lines for branching Brownian motion. Ann. Probab.19 1195–1205. · Zbl 0738.60079
[7] Hering, H. (1971). Critical Markov branching processes with general set of types. Trans. Amer. Math. Soc.160 185–202. · Zbl 0232.60067
[8] Kahane, J.-P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math.22 131–145. · Zbl 0349.60051
[9] Kallenberg, O. (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling77. Springer, Cham. · Zbl 1376.60003
[10] Kyprianou, A. E. (1999). A note on branching Lévy processes. Stochastic Process. Appl.82 1–14. · Zbl 0997.60090
[11] Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd ed. Cambridge Univ. Press, Cambridge. · Zbl 1165.60001
[12] Peyrière, J. (1974). Turbulence et dimension de Hausdorff. C. R. Acad. Sci. Paris Sér. A278 567–569.
[13] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics68. Cambridge Univ. Press, Cambridge.
[14] Shi, Z. (2015). Branching Random Walks. Lecture Notes in Math.2151. Springer, Cham.
[15] Uchiyama, K. (1982). Spatial growth of a branching process of particles living in \({\mathbf{R}}^{d}\). Ann. Probab.10 896–918. · Zbl 0499.60088
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.