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

arXiv: https://arxiv.org/abs/2312.10322 · HTML: https://mlsanity.com/paper/2312.10322/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Dec 2023.

## Built on by (most cited first, 1 of 1)

- [Stochastic Control Methods for Optimization](https://mlsanity.com/paper/2601.01248/) (2026)

## Abstract (arXiv)

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 solutions 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, Ekren, and Zhang [Proc. Amer. Math. Soc. (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.