Emergence of Common Noise: Quantitative Conditional Propagation of Chaos

We study a dynamical invariance principle for interacting particle systems with mean-field interactions and common noise emerging through a collective stochastic perturbation. The particle dynamics combine autonomous evolution with a weakly scaled random bombardment whose cumulative effect generates a Brownian common noise in the large population limit. Working in a general abstract framework based on stochastic flows and the stochastic sewing lemma, we establish quantitative conditional propagation of chaos estimates for both discrete Euler schemes and their continuous-time flow limits. Our approach yields explicit Wasserstein convergence rates and applies in particular to jump-diffusion models motivated by interacting spiking neuron systems. The analysis relies on quantitative central limit theorems of Rio and Bonis together with a stochastic sewing argument adapted to $L^2$-valued flows.