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

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mu\in\mathcal{P}(X)$, let $n_{a}:X\to\mathbb{R}$ be the Borel function of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel, let $\mathrm{id}$ be the identity map of $X$, and let $L^{2}(\mu;X^{a})$ be the space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, with norm $\lVert\cdot\rVert_{\mu}$; sequences and limits of sequences of real numbers are those of those definitions. A noise displacement for $\mu$ is a Borel map $v:X\to X$ with $v(x)\in X^{a}$ for every $x\in X$ and $\int_{X}n_{a}(v(x))\,\mu(dx)<\infty$; it is measurable as a map into $X^{a}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable and square-integrable with respect to $\mu$, since $|v(x)|_{a}^{2}=n_{a}(v(x))$ for every $x\in X$, and its class in $L^{2}(\mu;X^{a})$ is again written $v$. For such $v$ the map $\mathrm{id}+v$ is the composite of the map $(\mathrm{id},v)$, Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing since $\mathrm{id}$, being continuous, is Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, with the addition $X\times X\to X$, which is continuous by Properties of the Product of Two Real Inner Product Spaces §componentwise, The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset; hence $\mathrm{id}+v$ is Borel by claims 3 and 4 of Borel Measurability and Bounded Integration on a Metric Space; so $(\mathrm{id},\mathrm{id}+v):X\to X\times X$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing, and the push-forward $(\mathrm{id},\mathrm{id}+v)_{\#}\mu\in\mathcal{P}(X\times X)$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward is defined.

(Strong convergence of displacements)¶ Let $v$ be a noise displacement for $\mu$ and let $(v_{j})_{j\in\mathbb{N}}$ be a sequence of noise displacements for $\mu$. Suppose that

$$(\mathrm{id},\mathrm{id}+v_{j})_{\#}\mu\Rightarrow(\mathrm{id},\mathrm{id}+v)_{\#}\mu ,$$

and that for every positive $\varepsilon\in\mathbb{R}$ there is $J\in\mathbb{N}$ with $\lVert v_{j}\rVert_{\mu}\le\lVert v\rVert_{\mu}+\varepsilon$ for every $j\in\mathbb{N}$ with $J\le j$. Then $\lim_{j\to\infty}\lVert v_{j}-v\rVert_{\mu}=0$.