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 $f^{\text{'}}/2$.

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