Branching processes with competition and generalized Ray Knight Theorem

Abstract

We consider a discrete model of population with interaction where the birth and death rates are non linear functions of the population size. After proceeding to renormalization of the model parameters, we obtain in the limit of large population that the population size evolves as a diffusion solution of the SDE Z^x_t =x+\int_0^t f(Z^x_s)ds+2\int_0^t\int_0^{Z^x_s}W(ds,du), where W(ds,du) 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'/2.