Viscosity Subsolution, Supersolution and Solution on the Lift of the Wasserstein Space

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: 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 on the lift over $\mathcal{D}_{\Sigma}$ with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair for each positive $\delta\in\mathbb{R}$, and let $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$. The space $L^{2}(\Omega;\mathbb{R}^{d})$ with its norm $\lVert\cdot\rVert_{L^{2}}$ and metric $d_{L^{2}}$ and the law $\mathcal{L}(X)$ of a class are those of that clause, and $\mathcal{D}^{\Lambda}$ and $\mathcal{D}_{\Sigma}^{\Lambda}$ are the preimages of $\mathcal{D}$ and of $\mathcal{D}_{\Sigma}$ under the law map $\Lambda$, so that $\mathcal{D}_{\Sigma}^{\Lambda}\subseteq\mathcal{D}^{\Lambda}$ because $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i). Since $(\Omega,\mathcal{F},P)$ is rich, every $\mu\in\mathcal{D}$ is the law of some $X\in\mathcal{D}^{\Lambda}$ by The Wasserstein Distance and the Mean-Square Distance of Random Vectors (iii); in particular $\mathcal{D}_{\Sigma}^{\Lambda}$ is nonempty, $\mathcal{D}_{\Sigma}$ being nonempty by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (ii).

Lifted test functions $\Phi$ and their translation Hessians $H_{\Phi}(X)\in\mathcal{S}(d)$ are those of that definition, the gradient $D\Phi(X)\in L^{2}(\Omega;\mathbb{R}^{d})$ of $\Phi$ at $X$ is that of that clause, and $\lVert\cdot\rVert$ on $\mathcal{S}(d)$ is the norm fixed there. Local maxima and local minima of a function on $\mathcal{D}^{\Lambda}$ relative to $\mathcal{D}^{\Lambda}$ are understood in the metric space $(L^{2}(\Omega;\mathbb{R}^{d}),d_{L^{2}})$. The $\delta$-envelopes of $u$ relative to the penalty pair are functions on $\mathcal{D}$, and $\mathcal{L}(X)\in\mathcal{D}$ for $X\in\mathcal{D}^{\Lambda}$. Finally $|s|$ is the absolute value of a real number $s$.

1. (Viscosity subsolution on the lift)(i) Suppose that $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, so that for every positive $\delta\in\mathbb{R}$ the envelope $u^{-}_{\delta}$ is defined on $\mathcal{D}$ and $u^{-}_{\delta}(\mathcal{L}(X))$ is defined for $X\in\mathcal{D}^{\Lambda}$. The function $u$ is a viscosity subsolution of $F$ on the lift relative to the penalty pair if for every positive $\delta\in\mathbb{R}$, every lifted test function $\Phi$, every $\hat{X}\in\mathcal{D}^{\Lambda}$ at which the function $\mathcal{D}^{\Lambda}\to\mathbb{R}$ with value $u^{-}_{\delta}(\mathcal{L}(X))-\Phi(X)$ at $X$ has a local maximum relative to $\mathcal{D}^{\Lambda}$, and every positive $\varepsilon\in\mathbb{R}$, there exist $Z\in\mathcal{D}_{\Sigma}^{\Lambda}$, $V\in L^{2}(\Omega;\mathbb{R}^{d})$, $s\in\mathbb{R}$ and $\mathbb{X}\in\mathcal{S}(d)$ such that

$$\lVert Z-\hat{X}\rVert_{L^{2}}<\varepsilon,\qquad\bigl|u^{-}_{\delta}(\mathcal{L}(Z))-u^{-}_{\delta}(\mathcal{L}(\hat{X}))\bigr|<\varepsilon,\qquad\bigl|s-u^{-}_{\delta}(\mathcal{L}(\hat{X}))\bigr|<\varepsilon,$$ $$\lVert V-D\Phi(\hat{X})\rVert_{L^{2}}<\varepsilon,\qquad\bigl\lVert\mathbb{X}-H_{\Phi}(\hat{X})\bigr\rVert<\varepsilon,\qquad F^{-}_{\delta}(Z,s,V,\mathbb{X})\le\varepsilon .$$

2. (Viscosity supersolution on the lift)(ii) Suppose that $u$ is bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, so that for every positive $\delta\in\mathbb{R}$ the envelope $u^{+}_{\delta}$ is defined on $\mathcal{D}$ and $u^{+}_{\delta}(\mathcal{L}(X))$ is defined for $X\in\mathcal{D}^{\Lambda}$. The function $u$ is a viscosity supersolution of $F$ on the lift relative to the penalty pair if for every positive $\delta\in\mathbb{R}$, every lifted test function $\Phi$, every $\hat{X}\in\mathcal{D}^{\Lambda}$ at which the function $\mathcal{D}^{\Lambda}\to\mathbb{R}$ with value $u^{+}_{\delta}(\mathcal{L}(X))-\Phi(X)$ at $X$ has a local minimum relative to $\mathcal{D}^{\Lambda}$, and every positive $\varepsilon\in\mathbb{R}$, there exist $Z\in\mathcal{D}_{\Sigma}^{\Lambda}$, $V\in L^{2}(\Omega;\mathbb{R}^{d})$, $s\in\mathbb{R}$ and $\mathbb{X}\in\mathcal{S}(d)$ such that

$$\lVert Z-\hat{X}\rVert_{L^{2}}<\varepsilon,\qquad\bigl|u^{+}_{\delta}(\mathcal{L}(Z))-u^{+}_{\delta}(\mathcal{L}(\hat{X}))\bigr|<\varepsilon,\qquad\bigl|s-u^{+}_{\delta}(\mathcal{L}(\hat{X}))\bigr|<\varepsilon,$$ $$\lVert V-D\Phi(\hat{X})\rVert_{L^{2}}<\varepsilon,\qquad\bigl\lVert\mathbb{X}-H_{\Phi}(\hat{X})\bigr\rVert<\varepsilon,\qquad-\varepsilon\le F^{+}_{\delta}(Z,s,V,\mathbb{X}).$$

3. (Viscosity solution on the lift)(iii) Suppose that $u$ is bounded above near each point and bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$. The function $u$ is a viscosity solution of $F$ on the lift relative to the penalty pair if it is both a viscosity subsolution and a viscosity supersolution of $F$ on the lift relative to the penalty pair.