The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $F$ be a second-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 penalty pair. For $\nu\in\mathcal{D}_{\Sigma}$ the score $\Sigma(\nu)$ lies in $T_{\nu}$, hence in $L^{2}(\nu;\mathbb{R}^{d})$, by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i). Sequences of couplings of vanishing cost, and strong and weak convergence along them, are those of that definition; limits of sequences of real numbers are those of that definition, and convergence of a sequence in $\mathcal{S}(d)$ is convergence in the metric $d_{\mathcal{S}(d)}$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iii). In this definition the letters $r$ and $c$ denote real numbers; the dimension written $r$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation is not used.

1. (Sequences of test data converging along couplings with bounded score)(i) Let $R\in\mathbb{R}$ be positive, let $\xi_{n}=(\nu_{n},r_{n},q_{n},Y_{n})\in\mathcal{W}$ for every $n\in\mathbb{N}$, let $\xi=(\nu,r,q,Y)\in\mathcal{W}$, and let $\pi_{n}\in\Pi(\nu_{n},\nu)$ for every $n\in\mathbb{N}$. 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 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 $r$ and $(Y_{n})_{n\in\mathbb{N}}$ converges to $Y$.

2. (Shift semicontinuity at a level)(ii) 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)(iii) 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 $0<R$.