First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation

This setting fixes the standing notation used by results on first-order equations on the measures noise-connected to a reference measure, and on their viscosity solutions relative to a noise penalty pair. It is layered on Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, whose notation is in force throughout: the separable real Hilbert space $X$ with its fixed orthonormal basis, the noise weights $a$ with bound $\bar{a}$, the noise space $X^{a}$ with norm $|\cdot|_{a}$, the spaces $L^{2}(\mu;X^{a})$ and $L^{2}(\pi;X^{a})$ with inner products $\langle\cdot,\cdot\rangle_{\mu}$, $\langle\cdot,\cdot\rangle_{\pi}$ and norms $\lVert\cdot\rVert_{\mu}$, $\lVert\cdot\rVert_{\pi}$, and the reference measure $\rho$. It introduces no new concept and asserts nothing beyond what the references attached to it supply.

1. (The noise Wasserstein space)¶ $\mathcal{P}^{a}_{\rho}$ is the set of The Measures Noise-Connected to the Reference Measure §space and $W_{a}$ the noise Wasserstein distance; $(\mathcal{P}^{a}_{\rho},W_{a})$ is the metric space of 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 §metric, and the results 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 and Completeness of the Quadratic Wasserstein Space and of the Noise Wasserstein Space over a Hilbert Space are in force by reference. Sequences, their convergence, open balls, local maxima and local minima relative to a subset, and the semicontinuous envelopes of Upper and Lower Semicontinuous Envelopes of a Real-Valued Function are understood in this metric space unless a result says otherwise.

2. (Couplings and pairings)¶ For $\mu,\nu\in\mathcal{P}(X)$, $\Pi(\mu,\nu)$ is the set of couplings of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background, $\Pi^{a}(\mu,\nu)\subseteq\Pi(\mu,\nu)$ the set of couplings of finite noise cost and $I^{a}(\pi)$ the noise cost of $\pi\in\Pi^{a}(\mu,\nu)$. The noise displacement pairing $\mathcal{J}^{a}(\eta,\pi)$, the cross pairing $\mathcal{K}^{a}(q,\eta,\pi)$ and the discrepancy $\int_{X\times X}|q(x)-\eta(y)|_{a}^{2}\,\pi(dz)$ are those of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion, where $z\in X\times X$ is written $z=(x,y)$. Sequences of couplings of vanishing noise cost and strong and weak convergence of noise fields along them are those of Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost. The results Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost, Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion and Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs, with the gluing of two couplings, are in force by reference.

3. (Optimal couplings, optimal maps and the map property)¶ $\mathrm{id}$ is the identity map of $X$. Noise-optimal couplings, noise-optimal maps $T$ with their displacements $T-\mathrm{id}\in L^{2}(\mu;X^{a})$, uniquely noise-mapped ordered pairs, and the noise map property of a subset of $\mathcal{P}^{a}_{\rho}$ are those of the cited definitions, and The Noise-Optimal Map: Transport Cost, Stability Along Nearly Optimal Couplings and Stability Under Perturbation of the Source is in force by reference.

4. (Gradients and test functions)¶ For $\mu\in\mathcal{P}^{a}_{\rho}$, $T^{a}_{\mu}$ is the noise tangent space. That a function $\varphi:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ is differentiable along noise couplings at $\mu$, and its gradient along noise couplings $\nabla\varphi(\mu)\in L^{2}(\mu;X^{a})$, are those of that definition; noise intrinsic test functions on a subset $Q\subseteq\mathcal{P}^{a}_{\rho}$ are those of that definition, and Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions is in force by reference. $\mathcal{F}C^{1}_{b}(X)$ is the set of bounded $C^{1}$ cylindrical functions and $\nabla_{a}\psi$ the noise gradient of $\psi\in\mathcal{F}C^{1}_{b}(X)$, a map on $X$; the symbol $\nabla\varphi(\mu)$ applied to a function on $\mathcal{P}^{a}_{\rho}$ always denotes the gradient along noise couplings.

5. (Penalty pairs, operators and envelopes)¶ Noise penalty pairs $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ on $\mathcal{P}^{a}_{\rho}$ are those of that definition. For $Q\subseteq\mathcal{P}^{a}_{\rho}$, the bundle $\mathcal{V}^{a}(Q)$ of noise fields over $Q$ and first-order equation operators over $Q$, with values $F(\nu,r,q)$, are those of that definition, as are, relative to a noise penalty pair, the $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ of an operator over $\mathcal{D}_{\Sigma}$; for a function $u:\mathcal{D}\to\mathbb{R}$, penalty-subordinate growth from above and from below and the $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$, functions on $\mathcal{D}$, are those of those definitions. In a result adopting this setting the letter $q$ denotes a noise field and the letters $r$, $s$ and $c$ denote real numbers unless that result says otherwise.