The Score-Perturbation Bound for 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 $\sigma$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation §parameters and $H$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation §hamiltonian. For $\mu\in\mathcal{P}(\mathbb{T}^{d})$, $p,\zeta\in T_{\mu}$ and a real $\delta$, the field $p+\delta\zeta$ belongs to $T_{\mu}$ by The Torus Tangent Space: Linearity of the Periodic Calculus, Closed Subspace, and Representation of Bounded Functionals on Gradients §subspace, so $H(\mu,p+\delta\zeta)$ is defined.

¶The Hamiltonian $H$ satisfies the score-perturbation bound if there are a real number $\delta_{0}>0$ and, for every real number $R>0$, a real number $C_{R}\ge0$ such that

$$\bigl|H(\mu,p+\delta\zeta)-H(\mu,p)\bigr|\le\frac{\sigma^{2}\delta}{4}\,\lVert\zeta\rVert_{\mu}^{2}+C_{R}\,\delta$$

for every $\mu\in\mathcal{P}(\mathbb{T}^{d})$, all $p,\zeta\in T_{\mu}$ with $\lVert p\rVert_{\mu}\le R$, and every real $\delta$ with $0<\delta\le\delta_{0}$.