Control on Hilbert Space and Mean Field Control: the Common Noise Case
Alain Bensoussan, P. Jameson Graber, Phillip Yam
TL;DR
This work develops a comprehensive Hilbert-space approach to mean field control with common noise by lifting the Wasserstein-geometry of probability measures to a Hilbert space of square-integrable random variables. It derives smooth, classical solutions to the Bellman equation in this Hilbert setting, along with a full derivation of the master equation that governs the gradient flow of the value function. Central contributions include explicit first- and second-order functional derivatives with respect to the measure, a convexity/coercivity regime ensuring a unique optimal control, and detailed regularity results linking the value function, its derivatives, and the master equation under common-noise dynamics. The framework unifies dynamic programming, stochastic calculus, and variational derivatives in a robust infinite-dimensional setting, with potential implications for potential mean field control and related areas in mean field theory.
Abstract
The objective of this paper is to provide an equivalent of the theory developed in P.~Cardaliaguet, F.~Delarue, J.M.~Lasry, P.L.~Lions \cite{CDLL}, following the approach of control on Hilbert spaces introduced by the authors in \cite{BGY-2}. We include the common noise in this paper, so the alternative is now complete. Since we consider a control problem, our theory applies only to Mean field control and not to mean field games. The assumptions are adapted to guarantee a unique optimal control, so they insure that the cost functional is strictly convex and coercive.