Annealed limit for a diffusive disordered mean-field model with random jumps

We study a sequence of N-particle mean-field systems, each driven by N simple point processes Z^N,i in a random environment. Each Z^N,i has the same intensity (f(X_t-^N))_t and at every jump time of Z^N,i, the process X^N does a jump of height U_i/√N where the U_i are disordered centered random variables attached to each particle. We prove the convergence in distribution of X^N to some limit process X̄ that is solution to an SDE with a random environment given by a Gaussian variable, with a convergence speed for the finite-dimensional distributions. This Gaussian variable is created by a CLT as the limit of the patial sums of the U_i. To prove this result, we use a coupling for the classical CLT relying on the result of (Z. Wahrsch. Verw. Gebiete 34 (1976) 33–58), that allows to compare the conditional distributions of X^N and X̄ given the environment variables, with the same Markovian technics as the ones used in (Bernoulli 28 (2022) 125–149).