We consider conditional McKean-Vlasov processes that arise in the study of hydrodynamic limits of interacting diffusions on random regular graphs. We establish an H-theorem that characterizes the long-term behavior of these processes. Specifically, we show that a certain function related to the limit of certain renormalized relative entropies serves as a global Lyapunov function for the associated measure flow. We also provide counterexamples to highlight some subtleties in the approach and discuss some open problems.

This is joint work with Kevin Hu.