The Shift-Semicontinuity Condition for a First-Order Equation Operator on the Noise Wasserstein Space

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 $\mathcal{P}^{a}_{\rho}$ and let $F$ be a first-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair, with the set $\mathcal{W}$ of test data for $F$ and the notion of an $R$-bounded test datum, taken relative to this operator and this pair. For $\nu\in\mathcal{D}_{\Sigma}$ the score $\Sigma(\nu)$ lies in $T^{a}_{\nu}$, hence in $L^{2}(\nu;X^{a})$, by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair. Sequences of couplings of vanishing noise cost, and strong and weak convergence along them, are those of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §couplings; limits of sequences of real numbers are those of that definition.

1. (Sequences of test data converging along noise couplings with bounded score)¶ Let $R\in\mathbb{R}$ be positive, let $\xi_{n}=(\nu_{n},r_{n},q_{n})\in\mathcal{W}$ for every $n\in\mathbb{N}$, let $\xi=(\nu,r,q)\in\mathcal{W}$, and let $\pi_{n}\in\Pi^{a}(\nu_{n},\nu)$ for every $n\in\mathbb{N}$. Here $\nu_{n},\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{P}^{a}_{\rho}$, $q_{n},\Sigma(\nu_{n})\in L^{2}(\nu_{n};X^{a})$ and $q,\Sigma(\nu)\in L^{2}(\nu;X^{a})$, by The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §bundle and Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, so that the convergence notions below apply. The sequence $(\xi_{n})_{n\in\mathbb{N}}$ converges to $\xi$ along $(\pi_{n})_{n\in\mathbb{N}}$ with score bounded by $R$ if every $\xi_{n}$ is $R$-bounded, if

$$\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\ \le\ R\qquad(n\in\mathbb{N}),$$

if $(\pi_{n})_{n\in\mathbb{N}}$ is a sequence of couplings of vanishing noise cost from $(\nu_{n})_{n\in\mathbb{N}}$ to $\nu$, if $(q_{n})_{n\in\mathbb{N}}$ converges strongly to $q$ and $(\Sigma(\nu_{n}))_{n\in\mathbb{N}}$ converges weakly to $\Sigma(\nu)$ along $(\pi_{n})_{n\in\mathbb{N}}$, and if $(r_{n})_{n\in\mathbb{N}}$ converges to $r$.

2. (Shift semicontinuity at a level)¶ Let $\delta,R\in\mathbb{R}$ satisfy $0<\delta<1$ and $0<R$. The operator $F$ is shift-semicontinuous at $(\delta,R)$ if the following two implications hold for every sequence $(\xi_{n})_{n\in\mathbb{N}}$ of test data that converges, in the sense of clause 1, to a test datum $\xi\in\mathcal{W}$ along some sequence of couplings with score bounded by $R$, and for every $c\in\mathbb{R}$. First, if for every positive $\varepsilon\in\mathbb{R}$ there is $N\in\mathbb{N}$ such that $F^{-}_{\delta}(\xi_{n})\le c+\varepsilon$ for every $n\in\mathbb{N}$ with $N\le n$, then

$$F^{-}_{\delta}(\xi)\ \le\ c .$$

Secondly, if for every positive $\varepsilon\in\mathbb{R}$ there is $N\in\mathbb{N}$ such that $c-\varepsilon\le F^{+}_{\delta}(\xi_{n})$ for every $n\in\mathbb{N}$ with $N\le n$, then

$$c\ \le\ F^{+}_{\delta}(\xi).$$

3. (The shift-semicontinuity condition)¶ The operator $F$ satisfies the shift-semicontinuity condition if it is shift-semicontinuous at $(\delta,R)$ for all $\delta,R\in\mathbb{R}$ with $0<\delta<1$ and $0<R$.