Viscosity Sub- and Supersolutions on the Torus Wasserstein Space are Entropy-Penalised Viscosity Sub- and Supersolutions

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 and satisfies the score-perturbation bound. Semicontinuity refers to the metric space $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$.

1. (Subsolutions)¶ Every upper semicontinuous viscosity subsolution of $(\mathrm{E})$ is an entropy-penalised viscosity subsolution of $(\mathrm{E})$.

2. (Supersolutions)¶ Every lower semicontinuous viscosity supersolution of $(\mathrm{E})$ is an entropy-penalised viscosity supersolution of $(\mathrm{E})$.