Continuity Along Couplings of a Hamiltonian 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, with $H$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation §hamiltonian and convergence along couplings as defined there.

¶The Hamiltonian $H$ is continuous along couplings if, for every $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and $p\in T_{\mu}$ and all sequences $(\mu_{n})_{n\in\mathbb{N}}$ in $\mathcal{P}(\mathbb{T}^{d})$ and $(p_{n})_{n\in\mathbb{N}}$ with $p_{n}\in T_{\mu_{n}}$ such that $((\mu_{n},p_{n}))_{n\in\mathbb{N}}$ converges along couplings to $(\mu,p)$, the real sequence $(H(\mu_{n},p_{n}))_{n\in\mathbb{N}}$ converges to $H(\mu,p)$.