A large multi-agent system with noise both in position and control

In this work, we consider a multi-population system where the dynamics of each agent evolve according to a system of stochastic differential equations in a general functional setup, determined by the global state of the system. Each agent is associated with a probability measure, that assigns the label accounting for the population to which the agent belongs. We do not assume any prior knowledge of the label of a single agent, and we allow that it can change as a consequence of the interaction among the agents. Furthermore, the system is affected by noise both in the agent's position and labels. First, we study the well-posedness of such a system and then a mean-field limit, as the number of agents diverges, is investigated together with the analysis of the properties of the limit distribution both with Eulerian and Lagrangian perspectives. As an application, we consider a large network of interacting neurons with random synaptic weights, introducing resets in the dynamics.