Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation, consider the equation $(\mathrm{E})$ of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §score-form, whose left-hand side is used below, and let $u:\Sigma^{2}_{d}\to\mathbb{R}$. When the Hamiltonian $\mathcal{H}$ is taken to be a specific function $\Sigma^{2}_{2d}\to\mathbb{R}$, the notions below refer to $(\mathrm{E})$ with that Hamiltonian. $\rho$ and $\sigma$ are as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §parameters; $\mathcal{E}:\mathcal{D}\to\mathbb{R}$, $\mathcal{D}_{\Xi}$ and $\Xi$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy; $\mathcal{H}$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §hamiltonian; bounded plans, the momentum norm $|\cdot|_{\mathrm{mom}}$ and the plan pairing $\mathcal{J}$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §pairings; and $J^{+}$ and $J^{-}$ are the plan superjet and the plan subjet with slack $0$.

1. (Subsolution)¶ $u$ is a free-energy-penalised viscosity subsolution of $(\mathrm{E})$ if there are a real number $\delta_{0}>0$ and a nondecreasing function $\omega:[0,\infty)\to[0,\infty)$ with the following property. For every function $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, every real $\delta$ with $0<\delta\le\delta_{0}$, every $\mu\in\mathcal{D}$ such that

$$u\bigl(\kappa_{d}(\nu)\bigr)-\varphi\bigl(\kappa_{d}(\nu)\bigr)-\delta\,\mathcal{E}(\nu)<u\bigl(\kappa_{d}(\mu)\bigr)-\varphi\bigl(\kappa_{d}(\mu)\bigr)-\delta\,\mathcal{E}(\mu)\qquad\text{for every }\nu\in\mathcal{D}\text{ with }\nu\ne\mu,$$

and every bounded plan $\pi$ at $\mu$ with $\kappa_{2d}(\pi)\in J^{+}\varphi(\kappa_{d}(\mu))$, one has $\mu\in\mathcal{D}_{\Xi}$ and

$$\rho\,u\bigl(\kappa_{d}(\mu)\bigr)+\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)+\frac{\sigma^{2}}{2}\,\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)\le\delta\,\omega\bigl(|\pi|_{\mathrm{mom}}\bigr).$$

2. (Supersolution)¶ $u$ is a free-energy-penalised viscosity supersolution of $(\mathrm{E})$ if there are a real number $\delta_{0}>0$ and a nondecreasing function $\omega:[0,\infty)\to[0,\infty)$ with the following property. For every function $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, every real $\delta$ with $0<\delta\le\delta_{0}$, every $\mu\in\mathcal{D}$ such that

$$u\bigl(\kappa_{d}(\nu)\bigr)-\varphi\bigl(\kappa_{d}(\nu)\bigr)+\delta\,\mathcal{E}(\nu)>u\bigl(\kappa_{d}(\mu)\bigr)-\varphi\bigl(\kappa_{d}(\mu)\bigr)+\delta\,\mathcal{E}(\mu)\qquad\text{for every }\nu\in\mathcal{D}\text{ with }\nu\ne\mu,$$

and every bounded plan $\pi$ at $\mu$ with $\kappa_{2d}(\pi)\in J^{-}\varphi(\kappa_{d}(\mu))$, one has $\mu\in\mathcal{D}_{\Xi}$ and

$$\rho\,u\bigl(\kappa_{d}(\mu)\bigr)+\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)+\frac{\sigma^{2}}{2}\,\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)\ge-\delta\,\omega\bigl(|\pi|_{\mathrm{mom}}\bigr).$$

3. (Solution)¶ $u$ is a free-energy-penalised viscosity solution of $(\mathrm{E})$ if it is both a free-energy-penalised viscosity subsolution and a free-energy-penalised viscosity supersolution of $(\mathrm{E})$.