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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, assume that $(\Omega,\mathcal{F},P)$ is rich, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, 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, and let $u$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, with intrinsic gradient $\nabla u$ and translation Hessian $H_{u}$. The $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$, functions on $\mathcal{D}$, that a real-valued function is bounded above, or below, near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, the classical subsolutions, supersolutions and solutions of $F$ on a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$, the viscosity subsolutions, supersolutions and solutions of $F$ relative to the penalty pair, and lower semicontinuity are those of the definitions cited; local maxima and local minima relative to $\mathcal{D}$, and semicontinuity on $\mathcal{D}$ relative to $\mathcal{D}$, are taken in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §measures. For a test function $\chi$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and positive $\delta\in\mathbb{R}$, the functions $\chi-\delta\mathcal{E}$ and $\chi+\delta\mathcal{E}$ on $\mathcal{D}$ take the values $\chi(\mu)-\delta\,\mathcal{E}(\mu)$ and $\chi(\mu)+\delta\,\mathcal{E}(\mu)$ at $\mu$. Assume in addition:

(E1) $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$;

(E2) for every test function $\chi$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and every positive $\delta\in\mathbb{R}$, every point of $\mathcal{D}$ at which $\chi-\delta\mathcal{E}$ has a local maximum relative to $\mathcal{D}$ belongs to $\mathcal{D}_{\Sigma}$, and every point of $\mathcal{D}$ at which $\chi+\delta\mathcal{E}$ has a local minimum relative to $\mathcal{D}$ belongs to $\mathcal{D}_{\Sigma}$.

Then the following hold.

1. (Local bounds and envelopes)¶ The function $u$ is bounded above near each point and bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, and for every positive $\delta\in\mathbb{R}$ and every $\nu\in\mathcal{D}$,

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

3. (Subsolutions)¶ 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)¶ Suppose that $u$ is a classical supersolution of $F$ on $\mathcal{D}_{\Sigma}$. Let $\delta\in\mathbb{R}$ be positive, let $\varphi$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ 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

5. (Supersolutions)¶ 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)¶ 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.