Exact Inequalities at a Touching Point of a Penalised Viscosity Subsolution or Supersolution on the Wasserstein Space

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a Wasserstein-closed penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with closed score along couplings, and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair, that satisfies the shift-coercivity condition and the shift-semicontinuity condition. Viscosity subsolutions and supersolutions of $F$ relative to the pair, penalty-subordinate growth from above and from below, and the $\delta$-envelopes are those of the items cited; intrinsic test functions $\varphi$ on $\mathcal{D}$, their gradients along couplings $\nabla\varphi(\mu)$ at $\mu\in\mathcal{D}$ are those of the setting, and their translation Hessians $H_{\varphi}(\mu)$ are those of that definition; and local maxima and local minima relative to $\mathcal{D}$ are taken in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$. Let $\delta\in\mathbb{R}$ satisfy $0<\delta<1$ and let $\varphi$ be an intrinsic test function on $\mathcal{D}$.

1. (Subsolutions)¶ Let $u:\mathcal{D}\to\mathbb{R}$ have penalty-subordinate growth from above and be a viscosity subsolution of $F$ relative to the pair, and let $\hat{\mu}\in\mathcal{D}$ be a point at which the function $\mathcal{D}\to\mathbb{R}$ with value $u^{-}_{\delta}(\mu)-\varphi(\mu)$ at $\mu$ has a local maximum relative to $\mathcal{D}$. Then $\hat{\mu}\in\mathcal{D}_{\Sigma}$ and

$$F^{-}_{\delta}\bigl(\hat{\mu},\,u^{-}_{\delta}(\hat{\mu}),\,\nabla\varphi(\hat{\mu}),\,H_{\varphi}(\hat{\mu})\bigr)\le0 .$$

2. (Supersolutions)¶ Let $v:\mathcal{D}\to\mathbb{R}$ have penalty-subordinate growth from below and be a viscosity supersolution of $F$ relative to the pair, and let $\hat{\nu}\in\mathcal{D}$ be a point at which the function $\mathcal{D}\to\mathbb{R}$ with value $v^{+}_{\delta}(\nu)-\varphi(\nu)$ at $\nu$ has a local minimum relative to $\mathcal{D}$. Then $\hat{\nu}\in\mathcal{D}_{\Sigma}$ and

$$0\le F^{+}_{\delta}\bigl(\hat{\nu},\,v^{+}_{\delta}(\hat{\nu}),\,\nabla\varphi(\hat{\nu}),\,H_{\varphi}(\hat{\nu})\bigr).$$