## Electronic Journal of Probability

### Asymptotics of self-similar growth-fragmentation processes

#### Abstract

Markovian growth-fragmentation processes introduced in [8, 9] extend the pure-fragmentation model by allowing the fragments to grow larger or smaller between dislocation events. What becomes of the known asymptotic behaviors of self-similar pure fragmentations [6, 11, 12, 14] when growth is added to the fragments is a natural question that we investigate in this paper. Our results involve the terminal value of some additive martingales whose uniform integrability is an essential requirement. Dwelling first on the homogeneous case [8], we exploit the connection with branching random walks and in particular the martingale convergence of Biggins [18, 19] to derive precise asymptotic estimates. The self-similar case [9] is treated in a second part; under the so called Malthusian hypotheses and with the help of several martingale-flavored features recently developed in [10], we obtain limit theorems for empirical measures of the fragments.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 27, 30 pp.

Dates
Accepted: 6 March 2017
First available in Project Euclid: 21 March 2017

https://projecteuclid.org/euclid.ejp/1490061797

Digital Object Identifier
doi:10.1214/17-EJP45

Mathematical Reviews number (MathSciNet)
MR3629871

Zentralblatt MATH identifier
1358.60080

#### Citation

Dadoun, Benjamin. Asymptotics of self-similar growth-fragmentation processes. Electron. J. Probab. 22 (2017), paper no. 27, 30 pp. doi:10.1214/17-EJP45. https://projecteuclid.org/euclid.ejp/1490061797

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