Tamtam Proceedings - lamsin

530 El Saadiwhere δ Xi(t) is the Dirac measure at the location X i (t) ∈ R of the particle i at time t andN(t) is the total number of alive particles in the system at time t. For every B ∈ B(R),η t (B) counts the number of particles in B at time t.The infinitesimal generator of the interacting branching-diffusion process {η t } is denoted£ and is given by :£F ψ (ε) = 〈ε, Dψ′′ + 1 N (F a ∗ ε)ψ ′ + µ(Φ(ψ) − ψ)〉 exp〈ε, Logψ〉ψwhere Φ is the generating function of the offspring distribution (Φ(s) = 1 2 + 1 2 s2 ),F ψ (ε) = exp〈ε, Logψ〉 with ε ∈ M F (R) and ψ ∈ Cb 2 (R). The domain of £ is allsuch functions F ψ for which £F ψ is bounded.Now, we can state the main result on the characterization of the process {η t }.We recall:Definition 1 (Martingale Problem) We say that a stochastic process {ɛ(t)} t , or equivalentlyits distribution P π , solves the (£, π) martingale problem (where π ∈ M F (R)) if :∫ tP π [ɛ(0) = π] = 1 and F (ɛ(t)) − F (ɛ(0)) −is a P π -martingale for any function F ∈ D(£).0£F (ɛ(s))dsTheorem 2 (Martingale characterization)The distribution P ν of the process {η t } with initial measure ν is the unique solution to themartingale problem (£, ν).A proof of Theorem 2 is presented in [4].4. Weak convergence and limit of the system of particlesNow, we show that the empirical process {η t } when it is renormalized, converges to ameasure-valued process which is an extension of the Dawson-Watanabe superprocess.The idea to pass to the limit consists in applying the Feller rescaling [7]: the spatialmotion is left inchanged, but the number of particles, their mass and the branching rateare rescaled by considering that there are a very large number, N,of particles, each ofmass 1 1and of lifetimeN Nµ .Let {Y. (n) } n≥1 be the sequence of rescaled processes and consider at the n th stage aninitial measure consisting of N n particles each of whom is assigned a mass 1N nand hasan independant exponential lifetime of parameter µ n = N n µ, during which she movesaccording equation (1). At the end of her lifetime she dies and leaves behind, at the locationwhere she died, a random number of offspring (0 or 2) determined by the generatingfunction Φ. Suppose that the initial condition {Y (n)0 } n≥1 is convergent.TAMTAM –Tunis– 2005