, Limit theorems for some branching measure-valued processes, Advances in Applied Probability, vol.49, issue.2, pp.549-580, 2017.

, Limit theorems for some branching measure-valued processes, Advances in Applied Probability, vol.49, issue.2, pp.549-580, 2017.

, Limit theorems for some branching measure-valued processes, Advances in Applied Probability, vol.49, issue.2, pp.549-580, 2017.

, Limit theorems for some branching measure-valued processes, Advances in Applied Probability, vol.49, issue.2, pp.549-580, 2017.

, Limit theorems for some branching measure-valued processes, Advances in Applied Probability, vol.49, issue.2, pp.549-580, 2017.

, Limit theorems for some branching measure-valued processes, Advances in Applied Probability, vol.49, issue.2, pp.549-580, 2017.

, Limit theorems for some branching measure-valued processes, Advances in Applied Probability, vol.49, issue.2, pp.549-580, 2017.

, Limit theorems for some branching measure-valued processes, Advances in Applied Probability, vol.49, issue.2, pp.549-580, 2017.

, Limit theorems for some branching measure-valued processes, Advances in Applied Probability, vol.49, issue.2, pp.549-580, 2017.

, Limit theorems for some branching measure-valued processes, Advances in Applied Probability, vol.49, issue.2, pp.549-580, 2017.

, Limit theorems for some branching measure-valued processes, Advances in Applied Probability, vol.49, issue.2, pp.549-580, 2017.

R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, vol.65, 1975.

E. Aïdékon, J. Berestycki, E. Brunet, and Z. Shi, The branching brownian motion seen from its tip, 2011.

K. B. Athreya and P. E. Ney, Reprint of the 1972 original, vol.0373040, 2004.

D. Balagué, J. A. Cañizo, and P. Gabriel, Fine asymptotics of profiles and relaxation to equilibrium for growthfragmentation equations with variable drift rates, 2012.

V. Bansaye, J. Delmas, L. Marsalle, and V. Tran, Limit theorems for markov processes indexed by continuous time galton-watson trees, Annals of Applied Probability, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00431118

V. Bansaye and V. Tran, Branching feller diffusion for cell division with parasite infection, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00469680

J. Bertoin, On small masses in self-similar fragmentations, Stochastic Process. Appl, vol.109, issue.1, pp.13-22, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00101984

J. Bertoin, Random fragmentation and coagulation processes, Cambridge Studies in Advanced Mathematics, vol.102, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00103015

P. Billingsley, Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics, 1999.

D. Chafaï, F. Malrieu, and K. Paroux, On the long time behavior of the TCP window size process, Stochastic Processes and their Applications, pp.1518-1534, 2010.

J. Delmas and L. Marsalle, Detection of cellular aging in a Galton-Watson process, Stochastic Process. Appl, vol.120, issue.12, pp.2495-2519, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00293422

M. Doumic and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00408088

M. Doumic, M. Hoffmann, P. Reynaud-bouret, and V. Rivoirard, Nonparametric estimation of the division rate of a size-structured population, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00578694

M. Doumic, P. Maia, and J. Zubelli, On the calibration of a size-structured population model from experimental data, Acta Biotheoretica, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00412637

J. Engländer, S. C. Harris, and A. E. Kyprianou, Strong law of large numbers for branching diffusions, Ann. Inst. Henri Poincaré Probab. Stat, vol.46, issue.1, pp.279-298, 2010.

J. Engländer and A. Winter, Law of large numbers for a class of superdiffusions, Ann. Inst. H. Poincaré Probab. Statist, vol.42, issue.2, pp.171-185, 2006.

N. Fournier and S. Méléard, A microscopic probabilistic description of a locally regulated population and macroscopic approximations, Ann. Appl. Probab, vol.14, issue.4, pp.1880-1919, 2004.

I. Grigorescu and M. Kang, Steady state and scaling limit for a traffic congestion model, ESAIM Probab. Stat, vol.14, pp.271-285, 2010.

F. Guillemin, P. Robert, and B. Zwart, AIMD algorithms and exponential functionals, Ann. Appl. Probab, vol.14, issue.1, pp.90-117, 2004.
URL : https://hal.archives-ouvertes.fr/inria-00072141

S. C. Harris and D. Williams, Large deviations and martingales for a typed branching diffusion, I. Astérisque, issue.236, pp.133-154, 1996.

P. D. Hislop and I. M. Sigal, Introduction to spectral theory, Applied Mathematical Sciences, vol.113, 1996.

N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, vol.24, 1989.

J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes, of Grundlehren der Mathematischen Wissenschaften, vol.288

. Springer-verlag, , 2003.

A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes, Adv. in Appl. Probab, vol.18, issue.1, pp.20-65, 1986.

B. Jourdain, S. Méléard, and W. A. Woyczynski, Lévy flights in evolutionary ecology, 2011.

H. E. Kubitschek, Growth during the bacterial cell cycle: Analysis of cell size distribution, Biophysical Journal, 1969.

P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Advances in Applied Probability, vol.7, issue.2, pp.503-510, 2009.

A. H. Löpker and J. S. Van-leeuwaarden, Transient moments of the TCP window size process, J. Appl. Probab, vol.45, issue.1, pp.163-175, 2008.

S. Méléard, Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations, Stochastics Stochastics Rep, vol.63, issue.3-4, pp.195-225, 1998.

S. Méléard and S. Roelly, Sur les convergences étroite ou vague de processus à valeurs mesures, C. R. Acad. Sci. Paris Sér. I Math, vol.317, issue.8, pp.785-788, 1993.

S. Méléard and V. Tran, Slow and fast scales for superprocess limits of age-structured populations, 2010.

M. Métivier, Convergence faible et principe d'invariance pour des martingales à valeurs dans des espaces de Sobolev, Ann. Inst. H. Poincaré Probab. Statist, vol.20, issue.4, pp.329-348, 1984.

S. P. Meyn and R. L. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuoustime processes, Adv. in Appl. Probab, vol.25, issue.3, pp.518-548, 1993.

P. Michel, Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci, vol.16, issue.7, pp.1125-1153, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01485029

T. Ott, J. Kemperman, and M. Mathis, The stationary behavior of ideal tcp congestion avoidance, 1996.

B. Perthame, Transport equations in biology, Frontiers in Mathematics, 2007.

B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, vol.210, issue.1, pp.155-177, 2005.

R. G. Pinsky, Positive harmonic functions and diffusion, of Cambridge Studies in Advanced Mathematics, vol.45, 1995.

R. G. Pinsky, Positive harmonic functions and diffusion, of Cambridge Studies in Advanced Mathematics, vol.45, 1995.

S. T. Rachev, Probability metrics and the stability of stochastic models, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, 1991.

M. Reed and B. Simon, Methods of modern mathematical physics, IV. Analysis of operators, 1978.

S. Roelly-coppoletta, A criterion of convergence of measure-valued processes: application to measure branching processes, Stochastics, vol.17, issue.1-2, pp.43-65, 1986.

V. Tran, Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques, 2006.

C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics. American Mathematical Society, vol.58, 2003.

B. Cloez, C. Laboratoire-d'analyse-et-de-mathématiques-appliquées, U. Umr8050, and F. E. , -mail address: mailto:bertrand.cloez(at)univ-mlv