Open Problem—Weakly Interacting Particle Systems on Dense Random Graphs

We first briefly introduce the following classic mean-field particle system given as a collection of ${\mathbb{R}}^d$-valued stochastic processes {$X_i^n$(t) : t ∈ [0, T]${\}}_{i=1}^n$.

Here, {W_i} are independent and identically distributed standard d-dimensional Brownian motions, and ${\mathbb{R}}^d$ × ${\mathbb{R}}^d$ → ${\mathbb{R}}^d$ is assumed to be bounded and Lipschitz, but one could also allow for more general initial states, drift coefficient b, and a state-dependent diffusion coefficient.