Perron's Method on Unitary Laws: the Bump Construction

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, the test functions of Unitary Laws with Free Unitary Noise: Standing Data §test-functions, the constants $\rho,\beta$, the source $f$ and the noise direction $\widehat{\Theta}\lambda$, let $H$ be an admissible Hamiltonian. Let $w$ be a viscosity subsolution of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H$ and let $w_{*}$ be its lower semicontinuous envelope. Let $g:\mathcal{L}_{d}\to\mathbb{R}$ be lower semicontinuous on $(\mathcal{L}_{d},d_{\mathcal{L}})$ with $w(\lambda)\le g(\lambda)$ for every $\lambda\in\mathcal{L}_{d}$. Let $F\in C^{1}(E_{d})$ and $\lambda_{0}\in\mathcal{L}_{d}$ be such that $w_{*}-F\circ\iota_{d}$ has a local minimum at $\lambda_{0}$ relative to $\mathcal{L}_{d}$,

$$w_{*}(\lambda_{0})<g(\lambda_{0})\qquad\text{and}\qquad \rho\,w_{*}(\lambda_{0})+H\bigl(\lambda_{0},DF(\iota_{d}(\lambda_{0}))\bigr)-\frac{\beta^{2}}{2}\bigl\langle DF(\iota_{d}(\lambda_{0})),\widehat{\Theta}\lambda_{0}\bigr\rangle_{d}<f(\lambda_{0}).$$

(Bump)¶ For every real $r>0$ there is a viscosity subsolution $\widetilde{w}$ of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H$ such that $w(\lambda)\le\widetilde{w}(\lambda)\le g(\lambda)$ for every $\lambda\in\mathcal{L}_{d}$, $\widetilde{w}(\lambda)=w(\lambda)$ for every $\lambda\in\mathcal{L}_{d}$ with $r\le d_{\mathcal{L}}(\lambda,\lambda_{0})$, and $w(\lambda')<\widetilde{w}(\lambda')$ for some $\lambda'\in\mathcal{L}_{d}$ with $d_{\mathcal{L}}(\lambda',\lambda_{0})<r$.