Viscosity Solutions of a Class of Second Order Hamilton-Jacobi-Bellman Equations in the Wasserstein Space

This paper is devoted to solving a class of second order Hamilton-Jacobi-Bellman (HJB) equations in the Wasserstein space, associated with mean field control problems involving common noise. The well-posedness of viscosity solution to the HJB equation under a new notion is established under general assumptions on the coefficients. Our approach adopts the smooth metric developed by Bayraktar et al. (Proc Am Math Soc 151(09):4089-4098, 2023) as our gauge function for the purpose of smooth variational principle used in the proof of comparison theorem. Further estimates and regularity of the metric, including a novel second order derivative estimate with respect to the measure variable, are derived in order to ensure the uniqueness and existence.