Ergodic control of McKean-Vlasov systems on the Wasserstein space
Marco Fuhrman, Silvia Rudà
TL;DR
This work addresses ergodic control of McKean–Vlasov dynamics with coefficients depending on the marginal law by formulating the ergodic problem on the Wasserstein space. It develops a vanishing discount approach to construct a constant $\lambda$ and a Lipschitz potential $\phi$ on ${{\cal P}_2(\mathbb{R}^d)}$ that together solve a fully nonlinear elliptic ergodic HJB equation in viscosity sense, and shows how $\lambda$ governs the long-time behavior and Abelian–Tauberian limits of related finite-horizon problems. The study also develops a parabolic HJB theory on the Wasserstein space, proving the forward equation has a unique viscosity solution $w$ with long-time growth $w(t,\mu) \approx \phi(\mu)+\lambda t$ and linking finite-horizon and discounted formulations to $\lambda$; a verification framework yields optimal ergodic controls. Compared to prior work on Hilbert-space formulations, this paper treats the HJB on the Wasserstein space, providing complementary insights and a practical dynamic-programming route to ergodic mean-field control.
Abstract
We consider an optimal control problem with ergodic (long term average) reward for a McKean-Vlasov dynamics, where the coefficients of a controlled stochastic differential equation depend on the marginal law of the solution. Starting from the associated infinite time horizon expected discounted reward, we construct both the value $λ$ of the ergodic problem and an associated function $φ$, which provide a viscosity solution to an ergodic Hamilton-Jacobi-Bellman (HJB) equation of elliptic type. In contrast to previous results, we consider the function $φ$ and the HJB equation on the Wasserstein space, using concepts of derivatives with respect to probability measures. The pair $(λ,φ)$ also provides information on limit behavior of related optimization problems, for instance, results of Abelian-Tauberian type or limits of value functions of control problems for finite time horizon when the latter tends to infinity. Many arguments are simplified by the use of a functional relation for $φ$ in the form of a suitable dynamic programming principle.