Proof of Strong Convergence of Noise Displacement Fields from Weak Convergence of Their Graph Couplings and an Upper Bound on Their Norms

Each result cited is universally quantified over the data in its own statement. Elementary order and arithmetic of real numbers, including finite sums, squares, maxima and minima, is used through The Real Numbers: Standing Notation and Background §background without further citation.

Step 0 (Notation). By Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §noise-space and The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis, the vectors $f_{k}=a_{k}^{1/2}e_{k}$, $k\in\mathbb{N}$, form an orthonormal basis of the real Hilbert space $X^{a}$. By Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, $L^{2}(\mu;X^{a})$ is the space of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes taken with the measure space $(X,\mathcal{B}(X),\mu)$, with $E=X^{a}$ and with this basis, and its inner product $\langle\cdot,\cdot\rangle_{\mu}$ and norm $\lVert\cdot\rVert_{\mu}$ are those of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations; by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields it is a real Hilbert space, in particular a real inner product space whose norm is $\lVert\cdot\rVert_{\mu}$, so that $\langle u,u\rangle_{\mu}=\lVert u\rVert_{\mu}^{2}$ by Real Inner Product Space §norm. Let $L^{2}(\mu)=L^{2}(X,\mathcal{B}(X),\mu)$, a real inner product space with the inner product $\langle\cdot,\cdot\rangle_{L^{2}(\mu)}$ and norm $\lVert\cdot\rVert_{L^{2}(\mu)}$ of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product; by that clause, for $2$-integrable $g,h$ the product $gh$ is integrable and $\langle g,h\rangle_{L^{2}(\mu)}=\int_{X}gh\,d\mu$, classes being written like their representatives.

For a noise displacement $u$ for $\mu$ and $k\in\mathbb{N}$ let $u^{[k]}:X\to\mathbb{R}$ be the function $u^{[k]}(x)=\langle u(x),f_{k}\rangle_{a}$; it is the $k$-th coordinate of $u$ in the sense of Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality (written $u_{k}$ there; the bracket avoids a clash with the coordinates $x_{k}$ of points of $X$). It is Borel and $2$-integrable with respect to $\mu$ by Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §operations and Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §square-integrable, and its class in $L^{2}(\mu)$, again written $u^{[k]}$, is the $k$-th coordinate of the class $u\in L^{2}(\mu;X^{a})$ in the sense of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates. Since $u(x)\in X^{a}$, The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis gives

$$u^{[k]}(x)=a_{k}^{-1/2}\langle u(x),e_{k}\rangle\qquad(x\in X).\qquad(1)$$

By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates the series $\sum_{l=1}^{\infty}\lVert u^{[l]}\rVert_{L^{2}(\mu)}^{2}$ converges with sum $\lVert u\rVert_{\mu}^{2}$. Its terms are nonnegative, so its $k$-th term is at most its $k$-th partial sum, which is at most its sum by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates; with The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and Real Inner Product Space §norm this gives

$$\int_{X}\bigl(u^{[k]}\bigr)^{2}\,d\mu=\lVert u^{[k]}\rVert_{L^{2}(\mu)}^{2}\le\lVert u\rVert_{\mu}^{2}.\qquad(2)$$

Write $\gamma_{u}=(\mathrm{id},\mathrm{id}+u)_{\#}\mu\in\mathcal{P}(X\times X)$, defined as in the statement, and $\gamma_{j}=\gamma_{v_{j}}$, $\gamma=\gamma_{v}$; the hypothesis says $\gamma_{j}\Rightarrow\gamma$.

Step 1 (A common bound). By the hypothesis with $\varepsilon=1$ there is $J_{0}\in\mathbb{N}$ with $\lVert v_{j}\rVert_{\mu}\le\lVert v\rVert_{\mu}+1$ for $J_{0}\le j$. Put $B=\lVert v\rVert_{\mu}+1$; then $1\le B$, $\lVert v\rVert_{\mu}\le B$ and $\lVert v_{j}\rVert_{\mu}\le B$ for $J_{0}\le j$, so by (2), for every $k\in\mathbb{N}$,

$$\lVert v^{[k]}\rVert_{L^{2}(\mu)}^{2}\le B^{2},\qquad\lVert v_{j}^{[k]}\rVert_{L^{2}(\mu)}^{2}\le B^{2}\quad(J_{0}\le j).\qquad(3)$$

Step 2 (Convergence against one cylindrical coordinate). Fix $k\in\mathbb{N}$ and $\varphi\in\mathcal{F}C^{1}_{b}(X)$; its class lies in $L^{2}(\mu)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable. We show that $\bigl(\langle v_{j}^{[k]},\varphi\rangle_{L^{2}(\mu)}\bigr)_{j\in\mathbb{N}}$ converges to $\langle v^{[k]},\varphi\rangle_{L^{2}(\mu)}$.

By Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel, $\varphi$ is bounded; fix a bound $G$ for it as in Bounded Real-Valued Function on a Set. Moreover $\varphi$ is continuous: by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, $\varphi=\psi\circ p_{n}$ with $n\in\mathbb{N}$ and $\psi\in C^{1}_{b}(\mathbb{R}^{n})$; $\psi$ is of class $C^{1}$ on $\mathbb{R}^{n}$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, hence continuous on $\mathbb{R}^{n}$ as a map into $(\mathbb{R},d_{\mathbb{R}})$ by claims 1 and 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous; $p_{n}$ is Lipschitz with constant $1$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, hence continuous by A Lipschitz Map is Uniformly Continuous and Continuous; so $\varphi$ is continuous on $X$ by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map.

For positive $M\in\mathbb{R}$ let $c_{M}:\mathbb{R}\to\mathbb{R}$, $c_{M}(s)=\max\{-M,\min\{M,s\}\}$. Then $-M\le c_{M}(s)\le M$, and $c_{M}$ is the composite of $s\mapsto\min\{M,s\}$ and $t\mapsto\max\{-M,t\}$, each of which does not increase distances (elementary order arithmetic under The Real Numbers: Standing Notation and Background §background). Hence, for all $s,s'\in\mathbb{R}$,

$$|c_{M}(s)|\le M,\qquad|c_{M}(s)-c_{M}(s')|\le|s-s'|,\qquad|s-c_{M}(s)|\le\frac{s^{2}}{M};\qquad(4)$$

for the last, $c_{M}(s)=s$ if $|s|\le M$, and $|s-c_{M}(s)|=|s|-M<|s|<s^{2}/M$ if $M<|s|$. Define $F_{M}:X\times X\to\mathbb{R}$ by

$$F_{M}(z)=\varphi(\pi_{1}z)\,c_{M}\bigl(a_{k}^{-1/2}\langle\pi_{2}z-\pi_{1}z,e_{k}\rangle\bigr).$$

By (4), $|F_{M}(z)|\le GM$ for every $z$, so $F_{M}$ is bounded. It is continuous: let $z\in X\times X$ and let $(z_{m})_{m\in\mathbb{N}}$ converge to $z$ in $X\times X$. By Properties of the Product of Two Real Inner Product Spaces §componentwise, $X\times X$ being the product of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs, $(\pi_{i}z_{m})_{m}$ converges to $\pi_{i}z$ in $X$ for $i=1,2$. The coordinate function $x\mapsto\langle x,e_{k}\rangle$ is Lipschitz by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, hence continuous by A Lipschitz Map is Uniformly Continuous and Continuous. By continuity of $\varphi$ and of this function and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential, $(\varphi(\pi_{1}z_{m}))_{m}$ converges to $\varphi(\pi_{1}z)$ and $(\langle\pi_{i}z_{m},e_{k}\rangle)_{m}$ converges to $\langle\pi_{i}z,e_{k}\rangle$ for $i=1,2$, in $(\mathbb{R},d_{\mathbb{R}})$, equivalently in the sense of Limit of a Sequence of Real Numbers by The Real Numbers: Standing Notation and Background §sequences. Since $\langle\pi_{2}z-\pi_{1}z,e_{k}\rangle=\langle\pi_{2}z,e_{k}\rangle-\langle\pi_{1}z,e_{k}\rangle$ by Elementary Identities in a Real Inner Product Space §bilinear, and likewise for $z_{m}$, the numbers $t_{m}=a_{k}^{-1/2}\langle\pi_{2}z_{m}-\pi_{1}z_{m},e_{k}\rangle$ converge to $t=a_{k}^{-1/2}\langle\pi_{2}z-\pi_{1}z,e_{k}\rangle$ by Arithmetic of Limits of Real Sequences §scalar; then $(c_{M}(t_{m}))_{m}$ converges to $c_{M}(t)$ by (4), and $(F_{M}(z_{m}))_{m}$ converges to $F_{M}(z)$ by Arithmetic of Limits of Real Sequences §products. Hence $F_{M}$ is continuous on $X\times X$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset, and Borel by claims 2 and 3 of Borel Measurability and Bounded Integration on a Metric Space.

Let $u$ be a noise displacement for $\mu$ and $x\in X$. Then $\pi_{1}(x,x+u(x))=x$ and $\pi_{2}(x,x+u(x))-\pi_{1}(x,x+u(x))=u(x)$, so by (1)

$$F_{M}\bigl(x,x+u(x)\bigr)=\varphi(x)\,c_{M}\bigl(u^{[k]}(x)\bigr).$$

Since $F_{M}$ is bounded and Borel and $\gamma_{u}$ is a Borel probability measure on $X\times X$, $F_{M}$ is integrable with respect to $\gamma_{u}$ by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space; by claim 2 of Image Measures, Measures with Densities, and Change of Variables, through Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward, the function $\varphi\cdot(c_{M}\circ u^{[k]})$ is integrable with respect to $\mu$ and

$$\int_{X\times X}F_{M}\,d\gamma_{u}=\int_{X}\varphi(x)\,c_{M}\bigl(u^{[k]}(x)\bigr)\,\mu(dx).\qquad(5)$$

By Step 0, $u^{[k]}\varphi$ and $(u^{[k]})^{2}$ are integrable and $\langle u^{[k]},\varphi\rangle_{L^{2}(\mu)}=\int_{X}u^{[k]}\varphi\,d\mu$. The function $h_{u}=u^{[k]}\varphi-\varphi\cdot(c_{M}\circ u^{[k]})$ is integrable and, by (4), $|h_{u}(x)|\le G\,|u^{[k]}(x)-c_{M}(u^{[k]}(x))|\le\frac{G}{M}\,u^{[k]}(x)^{2}$ for every $x\in X$. By Linearity and Monotonicity of the Lebesgue Integral §integrable, (5) and (2),

$$\Bigl|\langle u^{[k]},\varphi\rangle_{L^{2}(\mu)}-\int_{X\times X}F_{M}\,d\gamma_{u}\Bigr|=\Bigl|\int_{X}h_{u}\,d\mu\Bigr|\le\int_{X}|h_{u}|\,d\mu\le\frac{G}{M}\,\lVert u^{[k]}\rVert_{L^{2}(\mu)}^{2}.\qquad(6)$$

Now let $\varepsilon>0$. First choose $M=1+3GB^{2}/\varepsilon$; it is positive and $GB^{2}/M<\varepsilon/3$, since $3GB^{2}<\varepsilon M$. Next, $F_{M}$ being bounded and continuous on $X\times X$ and $\gamma_{j}\Rightarrow\gamma$ in the sense of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §weak, Weak Convergence of Finite Borel Measures on a Metric Space and Limit of a Sequence of Real Numbers give $J_{1}\in\mathbb{N}$ with $\bigl|\int F_{M}\,d\gamma_{j}-\int F_{M}\,d\gamma\bigr|<\varepsilon/3$ for $J_{1}\le j$. For $j\ge\max\{J_{0},J_{1}\}$, (6) with $u=v_{j}$ and with $u=v$, together with (3), gives

$$\bigl|\langle v_{j}^{[k]},\varphi\rangle_{L^{2}(\mu)}-\langle v^{[k]},\varphi\rangle_{L^{2}(\mu)}\bigr|<\frac{GB^{2}}{M}+\frac{\varepsilon}{3}+\frac{GB^{2}}{M}<\varepsilon .$$

As $\varepsilon$ was arbitrary, the claim of this step follows by Limit of a Sequence of Real Numbers.

Step 3 (Cylindrical test fields). Let $m\in\mathbb{N}$ and $\varphi_{1},\dots,\varphi_{m}\in\mathcal{F}C^{1}_{b}(X)$. Let $h_{l}\in L^{2}(\mu)$ be the class of $\varphi_{l}$ for $l\le m$, which exists by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable, and the zero vector for $m<l$. The partial sums of $\sum_{l=1}^{\infty}\lVert h_{l}\rVert_{L^{2}(\mu)}^{2}$ are constant from index $m$ on, so the series converges by Series of Real Numbers §convergent, and The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §synthesis gives $w\in L^{2}(\mu;X^{a})$ whose $l$-th coordinate in the sense of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates is $h_{l}$ for every $l\in\mathbb{N}$; call $w$ the test field of $(\varphi_{1},\dots,\varphi_{m})$. For $u\in L^{2}(\mu;X^{a})$ with coordinates $u^{[l]}$, The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates gives $\langle u,w\rangle_{\mu}=\sum_{l=1}^{\infty}\langle u^{[l]},h_{l}\rangle_{L^{2}(\mu)}$, whose terms with $m<l$ vanish by Elementary Identities in a Real Inner Product Space §zero; the partial sums are therefore constant from index $m$ on, and

$$\langle u,w\rangle_{\mu}=\sum_{l=1}^{m}\langle u^{[l]},\varphi_{l}\rangle_{L^{2}(\mu)}.\qquad(7)$$

By (7), Step 2 applied with $(k,\varphi)=(l,\varphi_{l})$ for each $l\le m$, and Arithmetic of Limits of Real Sequences §sums applied $m-1$ times,

$$\lim_{j\to\infty}\langle v_{j},w\rangle_{\mu}=\langle v,w\rangle_{\mu}.\qquad(8)$$

Step 4 (Approximation of $v$ by test fields). We show: for every $\delta>0$ there is a test field $w$ with $\lVert v-w\rVert_{\mu}<\delta$. Let $\delta>0$; first $m$, then $\varphi_{1},\dots,\varphi_{m}$ are chosen. Let $s_{n}=\sum_{l=1}^{n}\lVert v^{[l]}\rVert_{L^{2}(\mu)}^{2}$. By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates and Series of Real Numbers §convergent, $(s_{n})_{n}$ converges to $\lVert v\rVert_{\mu}^{2}$, and $s_{n}\le\lVert v\rVert_{\mu}^{2}$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates; by Limit of a Sequence of Real Numbers choose $m\in\mathbb{N}$ with $\lVert v\rVert_{\mu}^{2}-s_{m}<\delta^{2}/2$. Next, the set of the classes of the functions in $\mathcal{F}C^{1}_{b}(X)$ is dense in $L^{2}(\mu)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §density, that is, by Real Hilbert Space §topology and Dense Subset of a Topological Space, its closure for the distance of Real Inner Product Space §distance is $L^{2}(\mu)$; so by Characterization of the Closure in a Metric Space by Open Balls (claim 1 implies claim 3) choose, for each $l\le m$, $\varphi_{l}\in\mathcal{F}C^{1}_{b}(X)$ with $\lVert v^{[l]}-\varphi_{l}\rVert_{L^{2}(\mu)}<\delta/(2m)$. Let $w$ be the test field of $(\varphi_{1},\dots,\varphi_{m})$. By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, applied to $v-w=v+(-1)w$, the $l$-th coordinate of $v-w$ is $v^{[l]}-\varphi_{l}$ for $l\le m$ and $v^{[l]}$ for $m<l$, and $\lVert v-w\rVert_{\mu}^{2}$ is the sum of the series of their squared norms in $L^{2}(\mu)$. For $m\le n$ the $n$-th partial sum of that series is

$$\sum_{l=1}^{m}\lVert v^{[l]}-\varphi_{l}\rVert_{L^{2}(\mu)}^{2}+(s_{n}-s_{m})<m\,\frac{\delta^{2}}{4m^{2}}+\bigl(\lVert v\rVert_{\mu}^{2}-s_{m}\bigr)<\frac{\delta^{2}}{4}+\frac{\delta^{2}}{2},$$

and for $n<m$ it is at most the $m$-th partial sum, the terms being nonnegative. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion the sum is the supremum of the partial sums, so $\lVert v-w\rVert_{\mu}^{2}\le3\delta^{2}/4<\delta^{2}$, and $\lVert v-w\rVert_{\mu}<\delta$.

Step 5 (Convergence of the inner products). We show $\lim_{j\to\infty}\langle v_{j},v\rangle_{\mu}=\lVert v\rVert_{\mu}^{2}$. Let $\varepsilon>0$; first put $\delta=\varepsilon/(3B)$, then choose by Step 4 a test field $w$ with $\lVert v-w\rVert_{\mu}<\delta$, then by (8) choose $J_{2}\in\mathbb{N}$ with $|\langle v_{j},w\rangle_{\mu}-\langle v,w\rangle_{\mu}|<\varepsilon/3$ for $J_{2}\le j$. Let $j\ge\max\{J_{0},J_{2}\}$. By Elementary Identities in a Real Inner Product Space §bilinear in $L^{2}(\mu;X^{a})$ and $\langle v,v\rangle_{\mu}=\lVert v\rVert_{\mu}^{2}$,

$$\langle v_{j},v\rangle_{\mu}-\lVert v\rVert_{\mu}^{2}=\langle v_{j},v-w\rangle_{\mu}+\bigl(\langle v_{j},w\rangle_{\mu}-\langle v,w\rangle_{\mu}\bigr)+\langle v,w-v\rangle_{\mu}.$$

By The Cauchy-Schwarz Inequality in a Real Inner Product Space in $L^{2}(\mu;X^{a})$, Elementary Identities in a Real Inner Product Space §homogeneity (giving $\lVert w-v\rVert_{\mu}=\lVert v-w\rVert_{\mu}$) and Step 1,

$$\bigl|\langle v_{j},v\rangle_{\mu}-\lVert v\rVert_{\mu}^{2}\bigr|<B\delta+\frac{\varepsilon}{3}+B\delta=\varepsilon .$$

The claim follows by Limit of a Sequence of Real Numbers.

Step 6 (Conclusion). Let $\varepsilon>0$. First put $\eta=\min\bigl\{1,\varepsilon^{2}/\bigl(2(2\lVert v\rVert_{\mu}+3)\bigr)\bigr\}$, which is positive; then choose by the hypothesis $J_{3}\in\mathbb{N}$ with $\lVert v_{j}\rVert_{\mu}\le\lVert v\rVert_{\mu}+\eta$ for $J_{3}\le j$, and by Step 5 $J_{4}\in\mathbb{N}$ with $|\langle v_{j},v\rangle_{\mu}-\lVert v\rVert_{\mu}^{2}|<\eta$ for $J_{4}\le j$. Let $j\ge\max\{J_{3},J_{4}\}$. Since $0\le\lVert v_{j}\rVert_{\mu}\le\lVert v\rVert_{\mu}+\eta$ and $\eta^{2}\le\eta$,

$$\lVert v_{j}\rVert_{\mu}^{2}\le\lVert v\rVert_{\mu}^{2}+2\eta\lVert v\rVert_{\mu}+\eta^{2}\le\lVert v\rVert_{\mu}^{2}+\eta\bigl(2\lVert v\rVert_{\mu}+1\bigr).$$

By Elementary Identities in a Real Inner Product Space §expansion in $L^{2}(\mu;X^{a})$,

$$\lVert v_{j}-v\rVert_{\mu}^{2}=\bigl(\lVert v_{j}\rVert_{\mu}^{2}-\lVert v\rVert_{\mu}^{2}\bigr)-2\bigl(\langle v_{j},v\rangle_{\mu}-\lVert v\rVert_{\mu}^{2}\bigr)<\eta\bigl(2\lVert v\rVert_{\mu}+1\bigr)+2\eta\le\frac{\varepsilon^{2}}{2}<\varepsilon^{2}.$$

As $0\le\lVert v_{j}-v\rVert_{\mu}$, this gives $\bigl|\lVert v_{j}-v\rVert_{\mu}-0\bigr|<\varepsilon$ for every $j\ge\max\{J_{3},J_{4}\}$. Since $\varepsilon$ was arbitrary, $\lim_{j\to\infty}\lVert v_{j}-v\rVert_{\mu}=0$ by Limit of a Sequence of Real Numbers.