Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation 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, consider the equation $(\mathrm{E})$ of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space §laplacian-form, and let $u:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$.

1. (Subsolution)¶ $u$ is a viscosity subsolution of $(\mathrm{E})$ if for every Laplacian test function $\varphi$ and every $\mu\in\mathcal{P}(\mathbb{T}^{d})$ with $u(\nu)-\varphi(\nu)\le u(\mu)-\varphi(\mu)$ for all $\nu\in\mathcal{P}(\mathbb{T}^{d})$,

$$\rho\,u(\mu)+H\bigl(\mu,\nabla\varphi(\mu)\bigr)-\frac{\sigma^{2}}{2}\,\mathcal{L}\varphi(\mu)\le0.$$

2. (Supersolution)¶ $u$ is a viscosity supersolution of $(\mathrm{E})$ if for every Laplacian test function $\varphi$ and every $\mu\in\mathcal{P}(\mathbb{T}^{d})$ with $u(\nu)-\varphi(\nu)\ge u(\mu)-\varphi(\mu)$ for all $\nu\in\mathcal{P}(\mathbb{T}^{d})$,

$$\rho\,u(\mu)+H\bigl(\mu,\nabla\varphi(\mu)\bigr)-\frac{\sigma^{2}}{2}\,\mathcal{L}\varphi(\mu)\ge0.$$

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