Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on the Wasserstein Space are Viscosity Sub- and Supersolutions

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with regular penalised maxima whose penalty $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$, and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$ that is degenerate elliptic, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair. Let $u$ be an intrinsic test function on $\mathcal{D}$ whose restriction to $\mathcal{D}$, again written $u$, has penalty-subordinate growth from above and from below, with $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$; it is also an intrinsic test function on $\mathcal{D}_{\Sigma}$ by Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space (i). Gradients along couplings $\nabla\varphi(\hat{\mu})$ and translation Hessians $H_{\varphi}(\hat{\mu})$ are those of The Intrinsic Calculus on the Wasserstein Space: Standing Notation (iii) and of that definition; classical subsolutions, supersolutions and solutions of $F$ on $\mathcal{D}_{\Sigma}$, and viscosity subsolutions, supersolutions and solutions of $F$ relative to the penalty pair, are those of the definitions cited; lower semicontinuity and local maxima and local minima relative to $\mathcal{D}$ are taken in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$. Then the following hold.

1. (Exact envelopes)(i) For every positive $\delta\in\mathbb{R}$ and every $\nu\in\mathcal{D}$,

$$u^{-}_{\delta}(\nu)=u(\nu)-\delta\,\mathcal{E}(\nu),\qquad u^{+}_{\delta}(\nu)=u(\nu)+\delta\,\mathcal{E}(\nu).$$

2. (Exact witnesses for subsolutions)(ii) Suppose that $u$ is a classical subsolution of $F$ on $\mathcal{D}_{\Sigma}$. Let $\delta\in\mathbb{R}$ be positive, let $\varphi$ be an intrinsic test function on $\mathcal{D}$ 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

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

3. (Subsolutions)(iii) If $u$ is a classical subsolution of $F$ on $\mathcal{D}_{\Sigma}$, then $u$ is a viscosity subsolution of $F$ relative to the penalty pair.

4. (Exact witnesses for supersolutions)(iv) Suppose that $u$ is a classical supersolution of $F$ on $\mathcal{D}_{\Sigma}$. Let $\delta\in\mathbb{R}$ be positive, let $\varphi$ be an intrinsic test function on $\mathcal{D}$ 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 minimum relative to $\mathcal{D}$. Then $\hat{\mu}\in\mathcal{D}_{\Sigma}$ and

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

5. (Supersolutions)(v) If $u$ is a classical supersolution of $F$ on $\mathcal{D}_{\Sigma}$, then $u$ is a viscosity supersolution of $F$ relative to the penalty pair.

6. (Solutions)(vi) If $u$ is a classical solution of $F$ on $\mathcal{D}_{\Sigma}$, then $u$ is a viscosity solution of $F$ relative to the penalty pair.