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

Each result cited below is universally quantified over the data in its own statement.

Write $W$ for $W_{\mathbb{T}}$ and $\mathcal{P}^{\mathrm{Ent}}$ for $\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$.

Step 1 (Continuous functions are semicontinuous). Let $w:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ be continuous, let $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and $\varepsilon>0$. By the definition of continuity for $W$ and the absolute-value metric there is a real $r>0$ such that $|w(\nu)-w(\mu)|<\varepsilon$, hence $w(\mu)-\varepsilon<w(\nu)<w(\mu)+\varepsilon$, for every $\nu$ with $W(\mu,\nu)<r$. So $w$ is upper semicontinuous and lower semicontinuous on $\mathcal{P}(\mathbb{T}^{d})$.

Step 2 (Comparison on finite-entropy measures). Let $u,v$ be as in clause 1. By Step 1, $u$ is upper semicontinuous and $v$ is lower semicontinuous. Since $H$ is continuous along couplings and satisfies the score-perturbation bound, Viscosity Sub- and Supersolutions on the Torus Wasserstein Space are Entropy-Penalised Viscosity Sub- and Supersolutions §sub shows that $u$ is an entropy-penalised viscosity subsolution of $(\mathrm{E})$, and Viscosity Sub- and Supersolutions on the Torus Wasserstein Space are Entropy-Penalised Viscosity Sub- and Supersolutions §super shows that $v$ is an entropy-penalised viscosity supersolution of $(\mathrm{E})$. Since $H$ also satisfies the structure condition, Comparison Principle for Entropy-Penalised Viscosity Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space §comparison gives $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{P}^{\mathrm{Ent}}$.

Step 3 (Extension to all measures, clause 1). Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and $\varepsilon>0$. By the continuity of $u$ and of $v$ at $\mu$, as unfolded in Step 1, and taking the smaller of the two radii, there is a real $r>0$ such that $|u(\nu)-u(\mu)|<\varepsilon$ and $|v(\nu)-v(\mu)|<\varepsilon$ for every $\nu$ with $W(\mu,\nu)<r$. Choose a real $s$ with $0<s\le\tfrac12$ and $d\,s<r^{2}$, and put $\nu=S_{s}\mu$, with $S_{s}$ the heat semigroup of Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation §conventions. Then $W(\nu,\mu)^{2}\le d\,s<r^{2}$ by Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information §distance, so $W(\mu,\nu)<r$ (symmetry of $W$, The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist §metric), and $\nu$ has finite entropy by Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information §entropy, that is $\nu\in\mathcal{P}^{\mathrm{Ent}}$ in the notation of The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets §finite-entropy-set. By Step 2,

$$u(\mu)<u(\nu)+\varepsilon\le v(\nu)+\varepsilon<v(\mu)+2\varepsilon .$$

As $\varepsilon>0$ was arbitrary, $u(\mu)\le v(\mu)$. This proves clause 1.

Step 4 (Uniqueness, clause 2). Let $u_{1},u_{2}$ be continuous viscosity solutions of $(\mathrm{E})$. Then $u_{1}$ is a viscosity subsolution and $u_{2}$ a viscosity supersolution, so $u_{1}\le u_{2}$ on $\mathcal{P}(\mathbb{T}^{d})$ by clause 1; exchanging the roles gives $u_{2}\le u_{1}$. Hence $u_{1}=u_{2}$, which proves clause 2.