Comparison Principle for Entropy-Penalised 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 $H$ satisfies the structure condition. Let $u,v:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$, where $u$ is upper semicontinuous and an entropy-penalised viscosity subsolution of $(\mathrm{E})$, and $v$ is lower semicontinuous and an entropy-penalised viscosity supersolution of $(\mathrm{E})$, semicontinuity referring to the metric space $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$.

¶Then $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$.