Stochastic monotone inclusion with closed loop distributions

In this paper, we study in a Hilbertian setting the first and second-order monotone inclusions related to stochastic optimization problems with decision-dependent distributions. The studied dynamics are formulated as monotone inclusions governed by Lipschitz perturbations of maximally monotone operators where the concept of equilibrium plays a central role. We discuss the relationship between the $ \mathbb{W}_1 $-Wasserstein Lipschitz behavior of the distribution and the so-called coarse Ricci curvature. As an application, we consider the monotone inclusions associated with stochastic optimization problems involving the sum of a smooth function with Lipschitz gradient, a proximable function, and a composite term.