Cameron–Martin translation formula for a separable Banach Gaussian

On the dense quotient $B^{\ast }\mathrm{/}N\subset H$, define $J([\mathrm{\ell }])=\mathrm{\ell }(\cdot )$ as an element of $L^2(\mu )$. The definition of the inner product makes $J$ an isometry, so it extends to an isometry $J:H\to L^2(\mu )$ by completeness [F6]. Let $z_h=J(u)$ and choose $[{\mathrm{\ell }}_n]\to u$ in the quotient completion; put $Z_n={\mathrm{\ell }}_n(W)$. Since $L^2(\mu )$ is formed from Borel-measurable functions on $(B,\mathcal{B}(B),\mu )$, choose a Borel-measurable representative $z_h$ of its class. Approximation in $L^2$ gives uniqueness up to $\mu$-null sets, with $\mathrm{\parallel }{Z}_n-z_h{\mathrm{\parallel }}_2\to 0$. For fixed ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_m\in B^{\ast }$, set $Y_n=(Z_n,{\mathrm{\ell }}_1(W),\dots ,{\mathrm{\ell }}_m(W))$ and $Y=(z_h,{\mathrm{\ell }}_1(W),\dots ,{\mathrm{\ell }}_m(W))$. Each $Y_n$ is centered multivariate Gaussian by [F2]. Cauchy–Schwarz [F5] gives $E\mathrm{\mid }{Z}_n-z_h\mathrm{\mid }\le \mathrm{\parallel }{Z}_n-z_h{\mathrm{\parallel }}_2\to 0$, $\mathrm{\mid }E[Z_n{\mathrm{\ell }}_j(W)]-E[z_h{\mathrm{\ell }}_j(W)]\mathrm{\mid }\le \mathrm{\parallel }{Z}_n-z_h{\mathrm{\parallel }}_2\mathrm{\parallel }{\mathrm{\ell }}_j(W){\mathrm{\parallel }}_2\to 0$, and $\mathrm{\mid }E[Z_n^2]-E[z_h^2]\mathrm{\mid }\le \mathrm{\parallel }{Z}_n-z_h{\mathrm{\parallel }}_2(\mathrm{\parallel }{Z}_n{\mathrm{\parallel }}_2+\mathrm{\parallel }{z}_h{\mathrm{\parallel }}_2)\to 0$. Thus their covariance matrices ${\mathrm{\Sigma }}_n$ converge entrywise to the positive-semidefinite covariance matrix $\mathrm{\Sigma }$ of $Y$. For each $t=(s,t_1,\dots ,t_m)$, $\mathrm{\mid }E{e}^{it\cdot Y_n}-E{e}^{it\cdot Y}\mathrm{\mid }\le \mathrm{\mid }s\mathrm{\mid }E\mathrm{\mid }{Z}_n-z_h\mathrm{\mid }\to 0$. By [F2], $E{e}^{it\cdot Y_n}=e^{-t^{\mathsf{T}}{\mathrm{\Sigma }}_{n}t\mathrm{/}2}$, so the characteristic function of $Y$ is $e^{-t^{\mathsf{T}}\mathrm{\Sigma }t\mathrm{/}2}$, the characteristic function of $N(0,\mathrm{\Sigma })$. Uniqueness in [F2] proves that $Y$ is jointly Gaussian. Finally, its mixed covariances satisfy $\mathrm{Cov}(z_h,{\mathrm{\ell }}_j(W))={\lim }_n⟨[{\mathrm{\ell }}_n],[{\mathrm{\ell }}_j]{⟩}_H=⟨u,[{\mathrm{\ell }}_j]{⟩}_H={\mathrm{\ell }}_j(h)$ by [F1].

Choose a countable norming family $({\mathrm{\ell }}_j{)}_{j\ge 1}$ for $B$ using [A1, A2], and put $D=\exp (z_h-\mathrm{\parallel }h{\mathrm{\parallel }}_H^2\mathrm{/}2)$. By [F2], for every finite subvector $L=({\mathrm{\ell }}_1(W),\dots ,{\mathrm{\ell }}_m(W))$, the joint vector $(z_h,L)$ is a linear image of a finite standard Gaussian vector. For one standard coordinate $G$ and real $u,v$, completing the square in [F4]'s density gives $E{e}^{(u+iv)G}=e^{u^2\mathrm{/}2}\int e^{ivy}(2\pi {)}^{-1\mathrm{/}2}{e}^{-(y-u{)}^2\mathrm{/}2}dy=e^{(u+iv{)}^2\mathrm{/}2}$. Multiplying these identities over the independent standard coordinates and using the covariances from step 1.1 gives $E_{\mu }[e^{it\cdot L}D]=\exp \text{\hspace{0.17em}⁣}(-\frac{1}{2}{t}^{\mathsf{T}}\mathrm{Cov}(L)t+i{\sum }_{j=1}^m{t}_j{\mathrm{\ell }}_j(h))\mathrm{.}$ At $t=0$ this also gives $E_{\mu }D=1$. Since $z_h$ is finite almost surely, $D>0$ almost surely, so $Dd\mu$ is a probability equivalent to $\mu$. The displayed function is the characteristic function of $L+({\mathrm{\ell }}_1(h),\dots ,{\mathrm{\ell }}_m(h))$. Thus $Dd\mu$ and the law ${\mu }_h$ of $W+h$ have the same finite-dimensional pushforwards under a countable norming family.

By [F3], equality of those pushforwards implies $Dd\mu ={\mu }_h$ on all Borel sets, proving the Radon–Nikodym identity. Positivity of $D$ gives equivalence. If $h=C\mathrm{\ell }$, then $u=[\mathrm{\ell }]$ and $J(u)=\mathrm{\ell }(\cdot )$; if $C\mathrm{\ell }=C{\mathrm{\ell }}^{\mathrm{\prime }}$, the well-definedness theorem gives $[\mathrm{\ell }]=[{\mathrm{\ell }}^{\mathrm{\prime }}]$, so both the scalar representative and its variance agree almost surely.