$\require{physics}\newcommand{\cbrt}[1]{\sqrt[3]{#1}}\newcommand{\sgn}{\text{sgn}}\newcommand{\ii}[1]{\textit{#1}}\newcommand{\eps}{\varepsilon}\newcommand{\EE}{\mathbb E}\newcommand{\PP}{\mathbb P}\newcommand{\Var}{\mathrm{Var}}\newcommand{\Cov}{\mathrm{Cov}}\newcommand{\pperp}{\perp\kern-6pt\perp}\newcommand{\LL}{\mathcal{L}}\newcommand{\pa}{\partial}\newcommand{\AAA}{\mathscr{A}}\newcommand{\BBB}{\mathscr{B}}\newcommand{\CCC}{\mathscr{C}}\newcommand{\DDD}{\mathscr{D}}\newcommand{\EEE}{\mathscr{E}}\newcommand{\FFF}{\mathscr{F}}\newcommand{\WFF}{\widetilde{\FFF}}\newcommand{\GGG}{\mathscr{G}}\newcommand{\HHH}{\mathscr{H}}\newcommand{\PPP}{\mathscr{P}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathbb{H}}\DeclareMathOperator{\ess}{ess}\newcommand{\CC}{\mathbb C}\newcommand{\FF}{\mathbb F}\newcommand{\NN}{\mathbb N}\newcommand{\QQ}{\mathbb Q}\newcommand{\RR}{\mathbb R}\newcommand{\ZZ}{\mathbb Z}\newcommand{\KK}{\mathbb K}\newcommand{\SSS}{\mathbb S}\newcommand{\II}{\mathbb I}\newcommand{\conj}[1]{\overline{#1}}\DeclareMathOperator{\cis}{cis}\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}\DeclareMathOperator*{\range}{range}\DeclareMathOperator*{\nul}{null}\DeclareMathOperator*{\Tr}{Tr}\DeclareMathOperator*{\tr}{Tr}\newcommand{\id}{1\!\!1}\newcommand{\Id}{1\!\!1}\newcommand{\der}{\ \mathrm {d}}\newcommand{\Zc}[1]{\ZZ / #1 \ZZ}\newcommand{\Zm}[1]{\left(\ZZ / #1 \ZZ\right)^\times}\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\End}{End}\newcommand{\GL}{\mathbb{GL}}\newcommand{\SL}{\mathbb{SL}}\newcommand{\SO}{\mathbb{SO}}\newcommand{\OO}{\mathbb{O}}\newcommand{\SU}{\mathbb{SU}}\newcommand{\U}{\mathbb{U}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\Cl}{\mathrm{Cl}}\newcommand{\gr}{\mathrm{gr}}\newcommand{\gl}{\mathfrak{gl}}\newcommand{\sl}{\mathfrak{sl}}\newcommand{\so}{\mathfrak{so}}\newcommand{\su}{\mathfrak{su}}\newcommand{\sp}{\mathfrak{sp}}\newcommand{\uu}{\mathfrak{u}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\hh}{\mathfrak{h}}\DeclareMathOperator{\Ad}{Ad}\DeclareMathOperator{\ad}{ad}\DeclareMathOperator{\Rad}{Rad}\DeclareMathOperator{\im}{im}\renewcommand{\BB}{\mathcal{B}}\newcommand{\HH}{\mathcal{H}}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Mat}{Mat}\DeclareMathOperator{\span}{span}\DeclareMathOperator{\proj}{proj}$ >[!definition] Gaussian White Noise >Let $(E,\GGG)$ be a measurable space. Let $\mu$ be a $\sigma$-finite measure. A ==**Gaussian White Noise**== with ==**intensity**== $\mu$ is an isometry $G$ from $L^2(E,\GGG,\mu)$ into a [[Gaussian Spaces|centered Gaussian space]] $H$:$\EE[G(f)G(g)] = \int fg d\mu,\qquad f,g\in H\subset L^2(E,\GGG,\mu).$ >[!example] >If $f = \id_A$ with $\mu(A) < \infty$, then $G(\id_A)\sim N(0, \mu(A))$. We write $G(A) = G(\id_A)$. > >If $A_1,\dots, A_n\in \GGG$ are finite and disjoint, $(G(A_i))_i$ is a Gaussian vector in $\RR^n$ with diagonal covariance matrix. > >If $A$ is finite, and a disjoint union of $A_1,A_2,\dots\in \GGG$, then $\id_A = \sum \id_{A_i}$, and $G(A) = \sum G(A_i)$ in $L^2(\Omega, \FFF,\PP)$. (In fact, the partial sums form a martingale, thus the series [[Martingale Convergence Theorems|converges AS]]). > [!proof]- Construction > Pick any orthonormal basis $\{f_i: i\in I\}$ of $L^2(E,\GGG,\mu)$, and pick some probability space $(\Omega, \FFF,\PP)$ on which we can construct a collection $(X_i)_{i\in I}$ of independent $N(0,1)$ RVs (see [[Ionescu-Tulcea]]). Then, let > $ > G(f) = \sum_{i\in I} \braket{f|f_i} X_i > $ > >> [!idea] >> Recall that [[hilbert|Parseval's identity]] holds regardless of whether your Hilbert space is separable. The summation above contains at most countably many nonzero terms, and is thus well-defined. > >[!idea] A Fake Measure >When we do things like try to construct the [[Weiner Integral]], we will often want to think of $G$ as a sort of measure on $(E,\GGG)$, which takes values in $H$. The above properties show why! Unfortunately, it is *not* true that $G(\bullet)(\omega)$ is a measure, in general. Let us learn the true story. >[!idea] Confusion point: the GWN for $\mu$ is not necessarily unique! >[!claim] Extracting the Intensity >Let $A\in \GGG$ be finite. Suppose there exists a sequence of partitions of $A$, $A = A_1^n\cup\dots\cup A_{k_n}^n$, whose [[mesh]] tends to $0$. Then, the sequence >$\lim_{n\to \infty} \sum_{j\leq k_n} G(A^n_j)^2 \stackrel{L^2}{\to} \mu(A).$ >[!proof]- Second Moment Method > Please observe that the LHS is a random variable, while the RHS is a constant! For any $n$, $\{G(A_i^n)\}_i$ are independent, and $\EE[G(A_j^n)^2] = \mu(A_j^n)$. Then$\EE\left[\left(\sum_{j\leq k_n} G(A^n_j)^2 - \mu(A)\right)^2\right] = 2\sum_{j\leq k_n} \mu(A^n_j)^2$because if $X\sim N(0,\sigma^2)$, $\Var(X^2) = 2\sigma^4$. The RHS vanishes because it is bounded by $\left(\sup_j \mu(A_j^n)\right)\mu(A)$.