The Structure Condition Along Optimal Maps 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 $H$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation §hamiltonian. Let $\mu,\nu\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$. Optimal maps from $\mu$ to $\nu$ and from $\nu$ to $\mu$ exist by McCann's Theorem on the Flat Torus: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map with a Periodic Potential §map. For an optimal map $T$ from $\mu$ to $\nu$, the function $\Phi_{\nu}=\tfrac12W_{\mathbb{T}}(\cdot,\nu)^{2}$ is an intrinsic test function on $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ with $\nabla\Phi_{\nu}(\mu)=-v_{T}$, by Half the Squared Torus Wasserstein Distance to a Fixed Measure is an Intrinsic Test Function on the Absolutely Continuous Measures §test, so $-v_{T}\in T_{\mu}$ by Intrinsic Test Functions on the Torus Wasserstein Space §differentiability; hence $-\alpha v_{T}\in T_{\mu}$ for every real $\alpha$, by The Torus Tangent Space: Linearity of the Periodic Calculus, Closed Subspace, and Representation of Bounded Functionals on Gradients §subspace. In the same way $\alpha v_{S}\in T_{\nu}$ for an optimal map $S$ from $\nu$ to $\mu$.

¶The Hamiltonian $H$ satisfies the structure condition if for every real number $\eta>0$ there is a real number $r>0$ such that

$$H(\nu,\alpha v_{S})-H(\mu,-\alpha v_{T})<\eta$$

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 $\alpha>0$ with $\alpha\,W_{\mathbb{T}}(\mu,\nu)^{2}+W_{\mathbb{T}}(\mu,\nu)<r$.