A Cross-Level Comparison Principle for First-Order Equations on a Fine Noise Wasserstein Space and a Family of Coarse Levels Linked by Mode Restrictions

In the setting of The Real Numbers: Standing Notation and Background, and of Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation for each pair of levels named below, the following data are given.

The fine level is given by data $X$, $\langle\cdot,\cdot\rangle$, $(e_{k})_{k\in\mathbb{N}}$, $a$, $\bar{a}$ and $\rho$ as in Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, and at the fine level the notation of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation is used without index. For every $N\in\mathbb{N}$, level $N$ is given by data $X^{(N)}$, $\langle\cdot,\cdot\rangle^{(N)}$, $(e^{(N)}_{k})_{k\in\mathbb{N}}$, $a^{(N)}$, $\bar{a}^{(N)}$ and $\rho^{(N)}$ of the same kind, and $(\kappa^{(N)},p^{(N)},j^{(N)})$ is a mode-restriction link from the fine level to level $N$, with mode restriction $p^{(N)}$ and mode embedding $j^{(N)}$; here, for each $N\in\mathbb{N}$, the fine level and level $N$ are read as level 1 and level 2 of Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §levels. Level-indexed notation of the cited items (Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §notation) that carries the index $1$, such as $W_{a,1}$, $\mathcal{P}^{a}_{\rho_{1}}$, $\nabla_{1}$ and $\lVert\cdot\rVert_{\mu,1}$, is read without index at the fine level, and notation that carries the index $2$ is read with the index $(N)$ at level $N$: $X^{(N)}_{a}$ is the noise space of $a^{(N)}$, $\lVert\cdot\rVert^{(N)}_{\nu}$ the norm of $L^{2}(\nu;X^{(N)}_{a})$, $\mathcal{P}^{a}_{\rho^{(N)}}$ the set of measures noise-connected to $\rho^{(N)}$, $W^{(N)}_{a}$ the noise Wasserstein distance and $T^{a,(N)}_{\nu}$ the noise tangent space at $\nu$, all taken at level $N$.

$\mathcal{P}=(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$, and $F$ is a first-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to $\mathcal{P}$. For every $N\in\mathbb{N}$, $\mathcal{P}^{(N)}=(\mathcal{D}^{(N)},\mathcal{D}^{(N)}_{\Sigma},\mathcal{E}^{(N)},\Sigma^{(N)})$ is a noise penalty pair on $\mathcal{P}^{a}_{\rho^{(N)}}$ at level $N$, and $F^{(N)}$ is a first-order equation operator over $\mathcal{D}^{(N)}_{\Sigma}$ at level $N$, with $\delta$-shifts $F^{(N),-}_{\delta}$ and $F^{(N),+}_{\delta}$ relative to $\mathcal{P}^{(N)}$. Properness constants, score bounds and structure pairs for $F^{(N)}$ are taken at level $N$, with $Q=\mathcal{D}^{(N)}_{\Sigma}$ and relative to $\mathcal{P}^{(N)}$ where the definition involves a pair; $|s|$ is the absolute value of $s\in\mathbb{R}$. The following hypotheses are made.

(H1) (Fine level)¶ $\mathcal{P}$ is noise-closed and has closed score along noise couplings, and $\mathcal{D}$ has the noise map property; $F$ satisfies the shift-coercivity condition and the shift-semicontinuity condition and has momentum-continuous shifts, all relative to $\mathcal{P}$.

(H2) (Coarse levels)¶ For every $N\in\mathbb{N}$, at level $N$: $\mathcal{P}^{(N)}$ is noise-closed and has closed score along noise couplings, $\mathcal{D}^{(N)}$ has the noise map property, and $F^{(N)}$ satisfies the shift-semicontinuity condition relative to $\mathcal{P}^{(N)}$.

(H3) (Uniform penalty bounds)¶ There is $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}^{(N)}(\nu)$ for every $N\in\mathbb{N}$ and every $\nu\in\mathcal{D}^{(N)}$, and for every $c\in\mathbb{R}$ there is $B_{c}\in\mathbb{R}$ with $W^{(N)}_{a}(\nu,\rho^{(N)})\le B_{c}$ for every $N\in\mathbb{N}$ and every $\nu\in\mathcal{D}^{(N)}$ with $\mathcal{E}^{(N)}(\nu)\le c$.

(H4) (Uniform properness)¶ For every positive $R\in\mathbb{R}$ there is a positive $\lambda\in\mathbb{R}$ that is a properness constant for $F^{(N)}$ at $R$ for every $N\in\mathbb{N}$.

(H5) (Uniform shift-coercivity)¶ For all $\delta,R\in\mathbb{R}$ with $0<\delta<1$ and $0<R$ there is a nonnegative $C\in\mathbb{R}$ that is a score bound for $F^{(N)}$ at $(\delta,R)$ for every $N\in\mathbb{N}$.

(H6) (Uniform structure)¶ For every positive $R\in\mathbb{R}$ there is a pair $(\omega_{1},\omega_{2})$ that is a structure pair for $F^{(N)}$ at $R$ for every $N\in\mathbb{N}$.

(H7) (Uniform momentum continuity)¶ For all $\delta,R,\eta\in\mathbb{R}$ with $0<\delta<1$, $0<R$ and $0<\eta$ there is a positive $\gamma\in\mathbb{R}$ such that, for every $N\in\mathbb{N}$,

$$\bigl|F^{(N),-}_{\delta}(\nu,r,q)-F^{(N),-}_{\delta}(\nu,r,q')\bigr|<\eta\qquad\text{and}\qquad\bigl|F^{(N),+}_{\delta}(\nu,r,q)-F^{(N),+}_{\delta}(\nu,r,q')\bigr|<\eta$$

for every $\nu\in\mathcal{D}^{(N)}_{\Sigma}$ with $\lVert\Sigma^{(N)}(\nu)\rVert^{(N)}_{\nu}\le R$, every $r\in\mathbb{R}$ with $|r|\le R$, and all $q,q'\in L^{2}(\nu;X^{(N)}_{a})$ with $\lVert q\rVert^{(N)}_{\nu}\le R$, $\lVert q'\rVert^{(N)}_{\nu}\le R$ and $\lVert q-q'\rVert^{(N)}_{\nu}<\gamma$.

(H8) (Uniform link)¶ There is $c_{0}\in\mathbb{R}$ with $0\le c_{0}$ such that for every $N\in\mathbb{N}$ one has $p^{(N)}_{\#}\rho\in\mathcal{P}^{a}_{\rho^{(N)}}$, and the pairs $\mathcal{P}$ and $\mathcal{P}^{(N)}$ are compatible with constant $c_{0}$.

(H9) (Vanishing consistency defect)¶ For all $\eta,\delta,R\in\mathbb{R}$ with $0<\eta$, $0<\delta<1$ and $0<R$ there is $N_{0}\in\mathbb{N}$ such that for every $N\in\mathbb{N}$ with $N_{0}\le N$ the fine level with $F$ and $\mathcal{P}$ and level $N$ with $F^{(N)}$ and $\mathcal{P}^{(N)}$, read as level 1 and level 2, are $\eta$-consistent at $(\delta,R)$.

A function on $\mathcal{D}$ that is bounded above, respectively below, has penalty-subordinate growth from above, respectively from below, relative to $\mathcal{P}$ by Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §growth, and likewise for functions on $\mathcal{D}^{(N)}$ at level $N$, the pairs being noise-closed; so the viscosity subsolutions and viscosity supersolutions below, of $F$ relative to $\mathcal{P}$ and of $F^{(N)}$ relative to $\mathcal{P}^{(N)}$ at level $N$, are defined. For $\mu\in\mathcal{D}$ one has $p^{(N)}_{\#}\mu\in\mathcal{D}^{(N)}$ by (H8). Then the following hold.

1. (Fine subsolutions below coarse supersolutions)¶ Let $b,b',\theta,c\in\mathbb{R}$ with $0<\theta$. There is $N_{1}\in\mathbb{N}$ such that for every $N\in\mathbb{N}$ with $N_{1}\le N$, every viscosity subsolution $u:\mathcal{D}\to\mathbb{R}$ of $F$ relative to $\mathcal{P}$ with $u(\mu)\le b$ for every $\mu\in\mathcal{D}$, and every viscosity supersolution $v:\mathcal{D}^{(N)}\to\mathbb{R}$ of $F^{(N)}$ relative to $\mathcal{P}^{(N)}$ with $b'\le v(\nu)$ for every $\nu\in\mathcal{D}^{(N)}$,

$$u(\mu)\le v(p^{(N)}_{\#}\mu)+\theta\qquad\text{for every }\mu\in\mathcal{D}\text{ with }\mathcal{E}(\mu)\le c.$$

2. (Coarse subsolutions below fine supersolutions)¶ Let $b,b',\theta,c\in\mathbb{R}$ with $0<\theta$. There is $N_{1}\in\mathbb{N}$ such that for every $N\in\mathbb{N}$ with $N_{1}\le N$, every viscosity subsolution $w:\mathcal{D}^{(N)}\to\mathbb{R}$ of $F^{(N)}$ relative to $\mathcal{P}^{(N)}$ with $w(\nu)\le b$ for every $\nu\in\mathcal{D}^{(N)}$, and every viscosity supersolution $z:\mathcal{D}\to\mathbb{R}$ of $F$ relative to $\mathcal{P}$ with $b'\le z(\mu)$ for every $\mu\in\mathcal{D}$,

$$w(p^{(N)}_{\#}\mu)\le z(\mu)+\theta\qquad\text{for every }\mu\in\mathcal{D}\text{ with }\mathcal{E}(\mu)\le c.$$