Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation 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 penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $F$ be a second-order equation operator 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 $u:\mathcal{D}\to\mathbb{R}$. Intrinsic test functions $\varphi$ on $\mathcal{D}$, their gradients along couplings $\nabla\varphi(\hat{\mu})\in L^{2}(\hat{\mu};\mathbb{R}^{d})$ at a point $\hat{\mu}$ of $\mathcal{D}$, which lie in $T_{\hat{\mu}}$ by property (b) of that definition, and their translation Hessians $H_{\varphi}(\hat{\mu})\in\mathcal{S}(d)$ are those of that definition. The set $\mathcal{D}$ contains $\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i), so that the $\delta$-envelopes below, functions on $\mathcal{D}$, are defined at every point of $\mathcal{D}_{\Sigma}$; local maxima and local minima of a function on $\mathcal{D}$ relative to $\mathcal{D}$ are understood in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i); the discrepancy $\int_{\mathbb{R}^{d+d}}\lVert q(x)-\eta(y)\rVert^{2}\,\pi(dz)$ of $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $\eta\in L^{2}(\mu;\mathbb{R}^{d})$ along $\pi\in\Pi(\nu,\mu)$ is the nonnegative real number fixed there, as is the cost $I(\pi)$; and $|s|$ is the absolute value of a real number $s$.

1. (Viscosity subsolution)(i) Suppose that $u$ has penalty-subordinate growth from above, so that for every positive $\delta\in\mathbb{R}$ the envelope $u^{-}_{\delta}$ is defined. The function $u$ is a viscosity subsolution of $F$ relative to the penalty pair if for every $\delta\in\mathbb{R}$ with $0<\delta<1$, every intrinsic test function $\varphi$ on $\mathcal{D}$, every $\hat{\mu}\in\mathcal{D}$ at which the function $\mathcal{D}\to\mathbb{R}$ with value $u^{-}_{\delta}(\mu)-\varphi(\mu)$ at $\mu$ has a local maximum relative to $\mathcal{D}$, and every positive $\varepsilon\in\mathbb{R}$, there exist $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi(\nu,\hat{\mu})$, $s\in\mathbb{R}$, $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $Y\in\mathcal{S}(d)$ such that

$$I(\pi)<\varepsilon^{2},\qquad\bigl|u^{-}_{\delta}(\nu)-u^{-}_{\delta}(\hat{\mu})\bigr|<\varepsilon,\qquad\bigl|s-u^{-}_{\delta}(\hat{\mu})\bigr|<\varepsilon,$$ $$\int_{\mathbb{R}^{d+d}}\bigl\lVert q(x)-\nabla\varphi(\hat{\mu})(y)\bigr\rVert^{2}\,\pi(dz)<\varepsilon^{2},\qquad\bigl\lVert Y-H_{\varphi}(\hat{\mu})\bigr\rVert<\varepsilon,\qquad F^{-}_{\delta}(\nu,s,q,Y)\le\varepsilon .$$

2. (Viscosity supersolution)(ii) Suppose that $u$ has penalty-subordinate growth from below, so that for every positive $\delta\in\mathbb{R}$ the envelope $u^{+}_{\delta}$ is defined. The function $u$ is a viscosity supersolution of $F$ relative to the penalty pair if for every $\delta\in\mathbb{R}$ with $0<\delta<1$, every intrinsic test function $\varphi$ on $\mathcal{D}$, every $\hat{\mu}\in\mathcal{D}$ at which the function $\mathcal{D}\to\mathbb{R}$ with value $u^{+}_{\delta}(\mu)-\varphi(\mu)$ at $\mu$ has a local minimum relative to $\mathcal{D}$, and every positive $\varepsilon\in\mathbb{R}$, there exist $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi(\nu,\hat{\mu})$, $s\in\mathbb{R}$, $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $Y\in\mathcal{S}(d)$ such that

$$I(\pi)<\varepsilon^{2},\qquad\bigl|u^{+}_{\delta}(\nu)-u^{+}_{\delta}(\hat{\mu})\bigr|<\varepsilon,\qquad\bigl|s-u^{+}_{\delta}(\hat{\mu})\bigr|<\varepsilon,$$ $$\int_{\mathbb{R}^{d+d}}\bigl\lVert q(x)-\nabla\varphi(\hat{\mu})(y)\bigr\rVert^{2}\,\pi(dz)<\varepsilon^{2},\qquad\bigl\lVert Y-H_{\varphi}(\hat{\mu})\bigr\rVert<\varepsilon,\qquad-\varepsilon\le F^{+}_{\delta}(\nu,s,q,Y) .$$

3. (Viscosity solution)(iii) Suppose that $u$ has penalty-subordinate growth from above and from below. The function $u$ is a viscosity solution of $F$ relative to the penalty pair if it is both a viscosity subsolution and a viscosity supersolution of $F$ relative to the penalty pair.