Conditional Comparison and Uniqueness for the Linear-Quadratic HJB Equation with Free Langevin Noise in a Wall, Penalised by Voiculescu's Free Entropy

In the setting of Free Products, Semicircular Laws, the Free Heat Flow, Free Fisher Information and Free Entropy: Standing Notation, let $-\chi^{*}:\mathcal{D}_{\chi}\to\mathbb{R}$ be the map $\lambda\mapsto-\chi^{*}(\lambda)$, and assume that the pair $(\mathcal{D}_{\chi},-\chi^{*})$ satisfies the tangent inequality clause of a free entropy penalty, as conjectured in Displacement Convexity of Minus the Free Entropy: the Tangent Inequality along Optimal Couplings §tangent.

1. (Penalty)¶ $(\mathcal{D}_{\chi},-\chi^{*})$ is a free entropy penalty.

In claims 2 and 3 the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation is adopted with $(\mathcal{D}_{0},\mathcal{E}_{0})=(\mathcal{D}_{\chi},-\chi^{*})$, which is admissible by claim 1; $\mathcal{D}=\mathcal{D}_{\chi}\cap\mathcal{D}_{R}$ is the domain of the wall-confined free energy as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy. Let $f$, the affine data $b_{\mu}=(A(\mu),c(\mu))$ $(\mu\in\Sigma^{2}_{d})$ and $\mathcal{H}^{\mathrm{LQ}}$ be as in The Linear-Quadratic Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §data, and let $(\mathrm{E}^{\mathrm{LQ}})$ be the equation of The Linear-Quadratic Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §equation. Assume that there are reals $a\ge0$ and $L\ge0$ such that, for all $\mu,\nu\in\Sigma^{2}_{d}$ and $i,j\in[d]$,

$$|A(\mu)_{ij}|\le a,\qquad|c(\mu)_{i}|\le a,\qquad|A(\mu)_{ij}-A(\nu)_{ij}|\le L\,\widehat{W}_{2}(\mu,\nu),\qquad|c(\mu)_{i}-c(\nu)_{i}|\le L\,\widehat{W}_{2}(\mu,\nu),$$

and that $f$ is bounded and uniformly continuous on $\Sigma^{2}_{d}$ with the metric of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics.

2. (Comparison)¶ Let $u,v:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded; let $u$ be weak-star upper semicontinuous on bounded laws and a free-energy-penalised viscosity subsolution of $(\mathrm{E}^{\mathrm{LQ}})$, and let $v$ be weak-star lower semicontinuous on bounded laws and a free-energy-penalised viscosity supersolution of $(\mathrm{E}^{\mathrm{LQ}})$. Then $u(\kappa_{d}(\mu))\le v(\kappa_{d}(\mu))$ for every $\mu\in\mathcal{D}$.

3. (Uniqueness)¶ Any two bounded functions $\Sigma^{2}_{d}\to\mathbb{R}$ that are weak-star continuous on bounded laws and free-energy-penalised viscosity solutions of $(\mathrm{E}^{\mathrm{LQ}})$ agree at $\kappa_{d}(\mu)$ for every $\mu\in\mathcal{D}$.