Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise-closed noise penalty pair on $\mathcal{P}^{a}_{\rho}$ with regular penalised maxima, let $\lambda_{0},\theta\in\mathbb{R}$ be positive, let $g:\mathcal{D}\to\mathbb{R}$, and let $F$ be the Hamilton-Jacobi operator with penalty drift of the pair with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$, with $\delta$-shifts relative to the pair. Viscosity subsolutions and supersolutions of $F$ relative to the pair are those of that definition, and penalty-subordinate growth from above and from below is that of that definition. Let $\kappa\in\mathbb{R}$, and let $u_{\kappa}:\mathcal{D}\to\mathbb{R}$ be the constant function with value $\kappa$. Then the following hold.

1. (Penalty-subordinate growth)¶ $u_{\kappa}$ has penalty-subordinate growth from above and from below.

2. (Constant subsolutions)¶ If $\lambda_{0}\,\kappa\le g(\nu)$ for every $\nu\in\mathcal{D}_{\Sigma}$, then $u_{\kappa}$ is a viscosity subsolution of $F$ relative to the pair.

3. (Constant supersolutions)¶ If $g(\nu)\le\lambda_{0}\,\kappa$ for every $\nu\in\mathcal{D}_{\Sigma}$, then $u_{\kappa}$ is a viscosity supersolution of $F$ relative to the pair.