Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho}$, let $F$ be a first-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 $u:\mathcal{D}\to\mathbb{R}$. Noise intrinsic test functions $\varphi$ on $\mathcal{D}$ and their gradients along noise couplings $\nabla\varphi(\hat{\mu})\in L^{2}(\hat{\mu};X^{a})$ at points $\hat{\mu}\in\mathcal{D}$, which exist by property (b) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §differentiability, are those of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §gradients. The set $\mathcal{D}$ contains $\mathcal{D}_{\Sigma}$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, 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 $(\mathcal{P}^{a}_{\rho},W_{a})$; for $\nu,\hat{\mu}\in\mathcal{P}^{a}_{\rho}$, $\pi\in\Pi^{a}(\nu,\hat{\mu})$, $q\in L^{2}(\nu;X^{a})$ and $\eta\in L^{2}(\hat{\mu};X^{a})$, the noise cost $I^{a}(\pi)$ and the discrepancy $\int_{X\times X}|q(x)-\eta(y)|_{a}^{2}\,\pi(dz)$ are the nonnegative real numbers of Couplings of Finite Noise Cost and Their Noise Cost §cost and Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy, the pair $(\nu,\hat{\mu})$ being noise-connected by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected; and $|s|$ is the absolute value of a real number $s$.

1. (Viscosity subsolution)¶ 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 noise penalty pair if for every $\delta\in\mathbb{R}$ with $0<\delta<1$, every noise 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 $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi^{a}(\nu,\hat{\mu})$, $s\in\mathbb{R}$ and $q\in L^{2}(\nu;X^{a})$ such that

$$I^{a}(\pi)<\varepsilon^{2},\qquad\bigl|u^{-}_{\delta}(\nu)-u^{-}_{\delta}(\hat{\mu})\bigr|<\varepsilon,\qquad\bigl|s-u^{-}_{\delta}(\hat{\mu})\bigr|<\varepsilon,$$ $$\int_{X\times X}\bigl|q(x)-\nabla\varphi(\hat{\mu})(y)\bigr|_{a}^{2}\,\pi(dz)<\varepsilon^{2},\qquad F^{-}_{\delta}(\nu,s,q)\le\varepsilon .$$

2. (Viscosity supersolution)¶ 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 noise penalty pair if for every $\delta\in\mathbb{R}$ with $0<\delta<1$, every noise 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 $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi^{a}(\nu,\hat{\mu})$, $s\in\mathbb{R}$ and $q\in L^{2}(\nu;X^{a})$ such that

$$I^{a}(\pi)<\varepsilon^{2},\qquad\bigl|u^{+}_{\delta}(\nu)-u^{+}_{\delta}(\hat{\mu})\bigr|<\varepsilon,\qquad\bigl|s-u^{+}_{\delta}(\hat{\mu})\bigr|<\varepsilon,$$ $$\int_{X\times X}\bigl|q(x)-\nabla\varphi(\hat{\mu})(y)\bigr|_{a}^{2}\,\pi(dz)<\varepsilon^{2},\qquad-\varepsilon\le F^{+}_{\delta}(\nu,s,q) .$$

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