Branching processes with interaction and a generalized Ray–Knight Theorem

Ba, Mamadou; Pardoux, Etienne

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1290-1313.

Nous considérons un modèle d’évolution d’une population avec interaction entre les individus, où les taux de naissance et de mort sont fonction de la taille de la population. Nous obtenons la limite en grande population après renormalisation, qui est solution de l’EDS

$$Z_t^x=x+{\int }_0^{t}f\left(Z_s^x\right)\mathrm{d}s+2{\int }_0^t{\int }_0^{Z_s^x}W(\mathrm{d}s,\mathrm{d}u),$$

où $W(\mathrm{d}s,\mathrm{d}u)$ est un bruit blanc sur ${[0,\infty )}^2$. Nous donnons une représentation de cette diffusion à la Ray–Knight, en fonction des temps locaux d’un mouvement brownien réfléchi $H$ avec une dérive qui dépend du temps local accumulé par $H$ à son niveau courant, à travers la fonction $f^{\text{'}}/2$.

We consider a discrete model of population dynamics with interaction between individuals, where the birth and death rates are nonlinear functions of the population size. We obtain the large population limit of a renormalization of our model as the solution of the SDE

$$Z_t^x=x+{\int }_0^{t}f\left(Z_s^x\right)\mathrm{d}s+2{\int }_0^t{\int }_0^{Z_s^x}W(\mathrm{d}s,\mathrm{d}u),$$

where $W(\mathrm{d}s,\mathrm{d}u)$ is a time space white noise on ${[0,\infty )}^2$. We give a Ray–Knight representation of this diffusion in terms of the local times of a reflected Brownian motion $H$ with a drift that depends upon the local time accumulated by $H$ at its current level, through the function $f^{\text{'}}/2$.

Mots clés : Galton–Watson processes with interaction, generalized Feller diffusion

[1] D. Aldous. Stopping times and tightness. Ann. Probab. 6 (1978) 335–340. | DOI | MR | Zbl

[2] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1999. | DOI | MR | Zbl

[3] D. A. Dawson and Z. Li. Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 (2012) 813–857. | DOI | MR | Zbl

[4] A. Joffe and M. Métivier. Weak convergence of sequences of semi-martingales with applications to multitype branching processes. Adv. in Appl. Probab. 18 (1986) 20–65. | DOI | MR | Zbl

[5] A. Lambert. The branching process with logistic growth. Ann. Probab. 15 (2005) 1506–1535. | MR | Zbl

[6] V. Le and E. Pardoux. Height and the total mass of the forest of genealogical trees of a large population with general competition. ESAIM Probab. Stat. 19 (2015) 172–193. | DOI | Numdam | MR | Zbl

[7] V. Le, E. Pardoux and A. Wakolbinger. Trees under attack: A Ray–Knight representation of Feller’s branching diffusion with logistic growth. Probab. Theory Related Fields 155 (2013) 583–619. | DOI | MR | Zbl

[8] J. R. Norris, L. C. G. Rogers and D. Williams. Self-avoiding random walks: A Brownian motion model with local time drift. Probab. Theory Related Fields 74 (1987) 271–287. | DOI | MR | Zbl

[9] E. Pardoux and A. Wakolbinger. From Brownian motion with a local time drift to Feller’s branching diffusion with logistic growth. Electron. Commun. Probab. 16 (2011) 720–731. | DOI | MR | Zbl

[10] E. Pardoux and A. Wakolbinger. A path-valued Markov process indexed by the ancestral mass. ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015) 193–212. | MR | Zbl

[11] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3d edition. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin, 1999. | DOI | MR | Zbl

[12] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften 233. Springer, Berlin, 1979. | MR | Zbl