Uniqueness of a Continuous Viscosity Solution of the Mean-Field Control Equation on the Torus

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation, let $(b,f)$ be a control datum, let the Hamiltonian be $H=H_{b,f}$, the control Hamiltonian of $(b,f)$, and consider the equation $(\mathrm{E}_{b,f})$ of The Discounted Hamilton-Jacobi-Bellman Equation of Mean-Field Control with Idiosyncratic Noise on the Torus §equation. Viscosity sub- and supersolutions and solutions are those of Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space with this Hamiltonian, and continuity of a function $\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ is continuity for $W_{\mathbb{T}}$ and the absolute-value metric of $\mathbb{R}$.

1. (Comparison)¶ Let $u,v:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ be continuous, with $u$ a viscosity subsolution and $v$ a viscosity supersolution of $(\mathrm{E}_{b,f})$. Then $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{P}(\mathbb{T}^{d})$.

2. (Uniqueness)¶ There is at most one continuous viscosity solution of $(\mathrm{E}_{b,f})$.