The Control Hamiltonian is Continuous Along Couplings and Satisfies the Score-Perturbation Bound and the Structure Condition

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation, with $\sigma$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation §parameters, let $(b,f)$ be a control datum with constant $L$, and let the Hamiltonian be $H=H_{b,f}$, the control Hamiltonian of $(b,f)$. Then the following hold.

1. (Continuity)¶ $H$ is continuous along couplings.

2. (Score perturbation)¶ $H$ satisfies the score-perturbation bound, with $\delta_{0}=\sigma^{2}/4$ and $C_{R}=2(R+L)^{2}/\sigma^{2}$ for every real $R>0$.

3. (Structure)¶ $H$ satisfies the structure condition; more precisely, $H(\nu,\alpha v_{S})-H(\mu,-\alpha v_{T})\le2L\bigl(\alpha W_{\mathbb{T}}(\mu,\nu)^{2}+W_{\mathbb{T}}(\mu,\nu)\bigr)$ for all $\mu,\nu\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, every optimal map $T$ from $\mu$ to $\nu$, every optimal map $S$ from $\nu$ to $\mu$ and every real $\alpha>0$.