Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space

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 $\mathcal{D}_{\Sigma}$ with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair for each positive $\delta\in\mathbb{R}$, and let $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$. Test functions $\varphi$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$, their intrinsic gradients $\nabla\varphi(\mu)\in L^{2}(\mu;\mathbb{R}^{d})$ and their translation Hessians $H_{\varphi}(\mu)\in\mathcal{S}(d)$ are those of that definition. In this definition test function always means a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\nabla\varphi(\mu)$ its intrinsic gradient, and the letter $q$ denotes a vector field; the test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test with their gradient maps $\nabla\psi$, and the dimension written $q$ in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces and in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §background, are not used. The couplings $\Pi(\nu,\mu)$ and their quadratic cost $I$, nonnegative, are those of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §wasserstein; for $\nu,\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\nu,\mu)$ the cost $I(\pi)$ is finite, hence a real number, by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite, and for $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $\eta\in L^{2}(\mu;\mathbb{R}^{d})$ the discrepancy $\int_{\mathbb{R}^{d+d}}\lVert q(x)-\eta(y)\rVert^{2}\,\pi(dz)$ of $q$ and $\eta$ along $\pi$ is the nonnegative real number of that clause, with $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$ for the coordinate projections of $\mathbb{R}^{d+d}$. The set $\mathcal{D}$ contains $\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, which is nonempty by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty, so that the $\delta$-envelopes below, defined 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 Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §measures; $\lVert\cdot\rVert$ on $\mathcal{S}(d)$ is the norm fixed there; and $|s|$ is the absolute value of a real number $s$.

1. (Viscosity subsolution)¶ 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 $\mathcal{D}$. The function $u$ is a viscosity subsolution of $F$ relative to the penalty pair if for every positive $\delta\in\mathbb{R}$, every test function $\varphi$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$, 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

2. (Viscosity supersolution)¶ 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 $\mathcal{D}$. The function $u$ is a viscosity supersolution of $F$ relative to the penalty pair if for every positive $\delta\in\mathbb{R}$, every test function $\varphi$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$, 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

3. (Viscosity solution)¶ Suppose that $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$. 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.