Lipschitz-Cone Comparison for the Quadratic Control Problem on Unitary Laws under Strong Free Noise

In the setting of Unitary Laws with Free Unitary Noise: Standing Data, with the conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces, let $H_{Q}$ be the quadratic control Hamiltonian, $\rho,\beta$ the constants, $f$ the source, and $K$ a trilinear constant. Suppose that there is a real $L_{f}\ge0$ with

$$|f(\mu)-f(\nu)|\le L_{f}\,d_{\mathcal{L}}(\mu,\nu)\quad\text{for all }\mu,\nu\in\mathcal{L}_{d},\qquad\text{and}\qquad L_{f}\le\frac{\rho\,\beta^{2}}{4K}.$$

1. (Lipschitz cone)¶ Let $L$ be a real number with $L_{f}/\rho\le L\le\beta^{2}/(4K)$. If $u$ is a viscosity subsolution and $v$ a viscosity supersolution of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H_{Q}$, then $u(\mu)-v(\nu)\le L\,d_{\mathcal{L}}(\mu,\nu)$ for all $\mu,\nu\in\mathcal{L}_{d}$.

2. (Comparison)¶ If $u$ and $v$ are as in clause 1, then $u(\lambda)\le v(\lambda)$ for every $\lambda\in\mathcal{L}_{d}$.

3. (Uniqueness)¶ The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H_{Q}$ has at most one viscosity solution.

4. (Lipschitz bound)¶ Every viscosity solution $V$ of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H_{Q}$ satisfies $|V(\mu)-V(\nu)|\le(L_{f}/\rho)\,d_{\mathcal{L}}(\mu,\nu)$ for all $\mu,\nu\in\mathcal{L}_{d}$.