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

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: 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 on the lift 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 the space $L^{2}(\nu;\mathbb{R}^{d})$ with its norm $\lVert\cdot\rVert_{\nu}$, and for $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\nu$ the composition $\Sigma(\nu)\circ X$ is the element of $L^{2}(\Omega;\mathbb{R}^{d})$ of that clause; here $\mathcal{L}(X)\in\mathcal{D}_{\Sigma}$ for $X\in\mathcal{D}_{\Sigma}^{\Lambda}$, the preimage of $\mathcal{D}_{\Sigma}$ under the law map. Convergence of a sequence in the space $L^{2}(\Omega;\mathbb{R}^{d})$ is convergence in its metric $d_{L^{2}}$, 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), and convergence of a sequence of real numbers is convergence in the metric of the absolute value, as fixed in Euclidean Space and Real Matrices: Standing Notation (v); weak convergence of a sequence in the real inner product space $L^{2}(\Omega;\mathbb{R}^{d})$ is that of that definition. 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 with bounded score)(i) Let $R\in\mathbb{R}$ be positive, let $\xi_{n}=(X_{n},r_{n},V_{n},\mathbb{X}_{n})\in\mathcal{W}$ for every $n\in\mathbb{N}$, and let $\xi=(X,r,V,\mathbb{X})\in\mathcal{W}$. The sequence whose $n$-th term is $\xi_{n}$ converges to $\xi$ with score bounded by $R$ if every $\xi_{n}$ is $R$-bounded, if

$$\bigl\lVert\Sigma(\mathcal{L}(X_{n}))\bigr\rVert_{\mathcal{L}(X_{n})}\ \le\ R\qquad(n\in\mathbb{N}),$$

if the sequences whose $n$-th terms are $X_{n}$, $V_{n}$, $r_{n}$ and $\mathbb{X}_{n}$ converge to $X$, to $V$, to $r$ and to $\mathbb{X}$ respectively, and if the sequence whose $n$-th term is $\Sigma(\mathcal{L}(X_{n}))\circ X_{n}$ converges weakly to $\Sigma(\mathcal{L}(X))\circ X$ in $L^{2}(\Omega;\mathbb{R}^{d})$.

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 of test data converging, in the sense of clause 1, to a test datum $\xi\in\mathcal{W}$ with score bounded by $R$, whose $n$-th term is written $\xi_{n}$, 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 $0<R$.