Comparison and Uniqueness for Continuous Viscosity Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation, consider the equation $(\mathrm{E})$ of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space §laplacian-form, and assume that the Hamiltonian $H$ is continuous along couplings, satisfies the score-perturbation bound and satisfies the structure condition. 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 $(\mathrm{E})$. 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 $(\mathrm{E})$.