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For an isonormal Gaussian process $X$ on a real Hilbert space $H$ and $k\in H$, the positive density $\exp (X(k)-\Vert k{\Vert }^2/2)$ has expectation one and defines an equivalent probability measure. Under this measure, $X(h)$ has mean $⟨h,k⟩$ and the same covariance as before. The shifted family $X(h)-⟨h,k⟩$ is again an isonormal Gaussian process. This follows by evaluating the joint Gaussian exponential formula for any finite collection of arguments. It is the Hilbert-space version of exponential tilting.

Because Gaussian white noise is not an ${\ell }^2$ vector, least squares over $\Theta \subset {\ell }^2$ is defined by maximizingover $\theta \in \Theta$, with the noise pairing interpreted through an isonormal Gaussian process. Differences of this contrast equal the formal differences of squared residual norms.

Observe $Y(t)={\int }_0^{t}f(s)ds+\sigma W(t)$ on $[0,1]$, where $f\in L^2[0,1]$ is an unknown deterministic drift and $W$ is standard Brownian motion. The observation against a deterministic test function is $Y(h)=⟨f,h⟩+\sigma W(h)$, with covariance ${\sigma }^2⟨h,g⟩$. Gaussian white noise is represented by an isonormal Gaussian process on test functions rather than a pointwise noise function. Taking $\sigma =n^{-1/2}$ is the usual statistical scaling.

Choose an orthonormal basis $(e_j)$ of the separable Hilbert space and independent standard normal random variables $(Z_j)$ on a common probability space. In finite dimension the sums below are finite; in dimension zero take $X(0)=0$. In infinite dimension defineIndependence, centring and unit variance giveBy the Parseval identity for a Hilbertian basis, the coefficient sequence is square-summable. Completeness of $L^2(\Omega )$ therefore supplies a limit, and we defineEvery partial-sum map is linear, and passage to the $L^2$ limit preserves this identity. Thus for each fixed $a,b,g,h$, $X(ag+bh)=aX(g)+bX(h)$ as an $L^2$ identity, hence an almost sure equality. This constructs an isonormal Gaussian process.

The usual meaning is almost-sure linearity for each fixed choice of arguments. A version linear simultaneously for every argument can also be chosen: select a Hamel basis of $H$, choose a measurable representative of $X$ on each basis vector, and extend each sample algebraically by finite sums. For every fixed $h$, this extension equals the constructed $L^2$ variable almost surely, so all its required distributions are unchanged.

Apply the construction of part (a) to the real Hilbert space $H=L^2([0,\infty ),du)$, and setChoose $X(0)=0$. The interval of length zero represents the zero vector of $H$, so $W_0=0$. All variables are defined on the single probability space already used for the isonormal Gaussian process.

For $0\le s<t$, linearity of the isonormal Gaussian process givesThe squared $L^2$ norm of this indicator is $t-s$, so part (a) yieldsAlso, the covariance formula gives $\mathrm{Cov}(W_s,W_t)=s\wedge t$, the Brownian covariance kernel. The Gaussian statement concerns the signed increment.

Let ${\Delta }_j=W_{t_j}-W_{t_{j-1}}$ for $1\le j\le n$. These increments are jointly normal, because each is obtained by evaluating the isonormal Gaussian process on an interval indicator. Indicators of distinct intervals are orthogonal in $L^2([0,\infty ))$, soBy uncorrelated jointly Gaussian variables are independent, the increments on all these disjoint intervals are independent. Reversing the sign of the first increment, as in the printed list, preserves this independence.

The three requested properties have now been obtained without an existence theorem for Brownian motion. One can also obtain continuous paths: the Gaussian fourth moment gives $\mathbb{E}|W_t-W_s{|}^4=3|t-s{|}^2$. The Kolmogorov continuity theorem therefore supplies a continuous modification on every finite time interval, which can be chosen consistently on the half-line. Modification preserves every finite-dimensional distribution and hence the independent Gaussian increments. This yields Brownian motion itself.

In the Gaussian white noise model, observe the entire pathwhere $W$ is standard Brownian motion and the deterministic drift belongs to L2 space on $[0,1]$. Equivalently, $dY(t)=f(t)dt+n^{-1/2}dW(t)$. For every deterministic $h\in L^2[0,1]$, the observed stochastic integral satisfiesThe noise $W(h)=\int hdW$ is an isonormal Gaussian process. In particular its variance is $\Vert h{\Vert }_2^2$. Gaussian white noise is interpreted through these integrals, rather than as an ordinary random function with a pointwise value at every time. No smoothness of $f$ is needed.

Here is the Gaussian maximum bound without independence. Put $M={\max }_{1\le i\le N}|g_i|$. It is integrable since it is bounded by ${\sum }_i|g_i|$. For every $\lambda >0$,The second inequality uses only the moment-generating function of the standard normal distribution, so independence is unnecessary. Jensen inequality gives $e^{\lambda EM}\le E{e}^{\lambda M}$, henceMinimizing at $\lambda =\sqrt{2\log (2N)}$ proves

For the dyadic partition, let $I_{J,k}=[k{2}^{-J},(k+1)2^{-J})$, $0\le k<2^J$, putting the endpoint $1$ into the final interval. The Haar scaling functionsform an orthonormal basis of the space $V_J$ of functions constant on each interval. The Haar wavelets can be written asFor $J\ge 1$, the constant function ${\varphi }_{0,0}=1$ together with ${\psi }_{j,k}$, $0\le j<J$, is another orthonormal basis of $V_J$. Indeed, each successive level splits a cell's two constants into their sum and difference; the number of basis elements is $1+{\sum }_{j=0}^{J-1}{2}^j=2^J$. Thus the Haar projection is the orthogonal projectionOn cell $I_{J,k}$ its value is $2^J{\int }_{I_{J,k}}f(t)dt$. The first formula also applies to $J=0$.

Estimate each coefficient by its observed stochastic integral:This is the Haar projection estimator in Gaussian white noise. The noise integrals over disjoint intervals form a Gaussian vector with zero off-diagonal covariance. The principle that uncorrelated jointly Gaussian variables are independent then makes these coefficients independent. ThereforeTaking expectations shows that ${\hat{\Pi }}_{V_J}f$ is an unbiased estimator of ${\Pi }_{V_J}f$, both coefficientwise and pointwise for the stated step-function representatives.

Only one scaling function is nonzero in each cell, so the supremum norm of the error has the exact formThe endpoint convention ensures the same identity at $1$. Apply the Gaussian maximum bound without independence with $N=2^J$ to obtain the supremum norm risk of a Haar projection estimator:The factor $2^{J/2}$ comes from cellwise scaling, while the extra square root of $J+1$ comes from taking the maximum over $2^J$ Gaussian errors.

Choose an orthonormal basis $(e_n)$ of the Dirichlet energy space and independent variables $({\xi }_n)$ with the standard normal distribution. The zero-boundary Gaussian free field is the formal seriesIts precise energy-indexed interpretation is the isonormal Gaussian processFor every fixed $f$, this sum converges in $L^2$ of the probability space by Parseval identity, and it is a centered Gaussian random variable. For $f,g\in H_0^1(D)$,This covariance characterizes its law independently of the chosen basis. The series does not define an $H_0^1$-valued random element: ${\sum }_n{\xi }_n^2=\infty$ almost surely. It is realized as a random distribution, and its pairing with ordinary test functions is constructed in part (b).

The expression is a Test-function pairing with a Gaussian free field, rather than the pointwise integral of an ordinary random function. For a real test function $\varphi$, take the Dirichlet solutionusing the Green-kernel identity allowed in the original PDF. In particular, $-\Delta f_{\varphi }=2\pi \varphi$. Integration by parts gives, initially for smooth compactly supported $u$ and then by continuity in the Dirichlet energy space,The integral functional is continuous in the energy norm: pull back to the unit disc, where the transformed test function has compact support, and use the Poincaré inequality. Hence $f_{\varphi }$ is its energy-space representer. DefineBy the isonormal Gaussian process definition, it has mean zero and varianceThe logarithmic singularity is locally integrable, so this variance is finite for smooth compactly supported test functions. ThusThe local TeX corrupted the double integral and the bracketed Green-kernel identity; the PDF supplies the identity above and does not assert an infinite variance.

Because white-noise observations are not themselves in ${\ell }^2$, define the least-squares estimator as a minimizer of the Gaussian least-squares contrast, equivalently a maximizer over $\theta \in \Theta$ ofwhere $W$ is the isonormal Gaussian process on ${\ell }^2$. The entropy assumption makes $W$ sample-continuous on compact $\Theta$, so a maximizer exists.

Put $\Delta =\hat{\theta }-{\theta }_0$. Comparison with ${\theta }_0$ gives the basic inequalityForthe entropy assumption and the Dudley entropy integral giveThe Borell-TIS inequality further gives

Set $r_n=c{n}^{-4/9}$. This is the balanceOn the shell $2^j{r}_n\le \Vert \theta -{\theta }_0{\Vert }_2<2^{j+1}{r}_n$, the basic inequality would requireFor $c$ sufficiently large, the expectation bound is at most half this threshold for every $j$. Borell concentration then bounds the shell probability bySumming the geometric sequence of shell bounds gives a quantity tending to zero, uniformly in ${\theta }_0\in \Theta$. Therefore

The notation $P_0=\mathcal{N}(0,I)$ denotes generalized Gaussian white noise over $H=L^2({\mathbb{R}}^2)$, not a Gaussian measure supported on $H$: the identity is not a trace-class operator there. For example, realize the observation on a suitable space of tempered distributions. Its Cameron-Martin space of a Gaussian measure is $H$. For deterministic $h\in H$, the stochastic pairing $W_m(h)$ has normal distribution $\mathcal{N}(0,\Vert h{\Vert }_H^2)$ under $P_0$ and is the isonormal Gaussian process indexed by $H$.

Interpret the stated maps between Sobolev spaces as bounded linear operators, as usual. In particular $K:H\to H^2({\mathbb{R}}^2)\subset H$ is bounded, so its adjoint operator $K^{\ast }$ is bounded on $H$. Hence $A=K^{\ast }K$ maps $H$ into $H$, which is the property needed here. The stronger smoothing follows by Sobolev duality: boundedness of $K:H^{-4}\to H^{-2}$ gives $K^{\ast }:H^2\to H^4$, and therefore $A:H\to H^4\subset H$. As $\Pi (H)=1$, the shift $Au$ belongs to the noise Cameron-Martin space of a Gaussian measure for $\Pi$-almost every $u$.

The white-noise likelihood for a square-integrable shift, obtained from the Cameron-Martin theorem for a Gaussian measure in its white-noise form, isHere $P_u$ is the law of $Au+\eta$. In an orthonormal basis $(e_j)$ of $H$, with white-noise coordinates $m_j$, the pairing under $P_0$ isFor fixed $u$ this has mean-square convergence and converges almost surely, since ${\sum }_j|⟨Au,e_j⟩{|}^2=\Vert Au{\Vert }_H^2$. The stipulated joint measurability allows the likelihood to be used under ${\nu }_0(du,dm)=\Pi (du)P_0(dm)$. One must not replace this expression by a finite $\Vert m-Au{\Vert }_H^2$, because white noise is not $H$-valued. On the unbounded domain ${\mathbb{R}}^2$, membership of white noise in a global unweighted negative Sobolev space must not be assumed either; the stochastic pairing avoids that issue.

Let $\nu (du,dm)=\Pi (du)P_u(dm)$ be the actual joint law. The shift formula gives $\nu \ll {\nu }_0$ with Radon-Nikodym derivative $L$. DefineFor each fixed admissible $u$, the moment-generating function of a normal distribution givesBy Tonelli theorem, $\int Z(m)P_0(dm)=1$, so $Z(m)<\infty$ for $P_0$-almost every $m$. The likelihood is finite and strictly positive for ${\nu }_0$-almost every $(u,m)$; Fubini's theorem then gives $Z(m)>0$ for $P_0$-almost every $m$. The marginal observation law is $Z(m)P_0(dm)$ and is an equivalent probability measure to $P_0$.

The Bayes formula for a dominated observation model therefore definesTo verify that it is the conditional distribution, for measurable sets $B$ of unknowns and $C$ of data,Thus the posterior distribution is well defined for almost every observation under the actual data law. This is the appropriate almost-everywhere assertion supplied by Bayes' theorem; the stated assumptions do not prescribe a canonical posterior at every exceptional generalized datum.

The zero-boundary Gaussian free field $h$ on $D$ is the isonormal Gaussian process over $H_0^1(D)$: for $f,g\in H_0^1(D)$, the variables $(h,f{)}_{\nabla }$ are centered jointly Gaussian and satisfyEquivalently, for an orthonormal basis $(e_n)$ of $H_0^1(D)$ and independent standard normal variables $({\xi }_n)$,as a random generalized function.

Generalized Gaussian white noise over a real Hilbert space $H=L^2$ is an isonormal Gaussian process $W(h)$ with covariance $⟨h,g{⟩}_H$. Its law $P_0$ is carried on a larger observation space; the identity is not the covariance of an infinite-dimensional $H$-valued Gaussian measure because it is not a trace-class operator. A deterministic signal $h\in H$ defines a translated observation law $P_h$ with Radon-Nikodym derivativeIn an orthonormal basis, $W_m(h)={\sum }_j{m}_j{h}_j$ is a stochastic series with total variance $\Vert h{\Vert }_H^2$, not an inner product of two $H$-valued observations. The formula follows from the Cameron-Martin theorem for a Gaussian measure in its generalized white-noise version, or from finite-dimensional Gaussian likelihood ratios. On ${\mathbb{R}}^2$, generalized noise can be realized on tempered distributions; an unweighted global negative Sobolev space is not automatically a suitable almost-sure support on an unbounded domain. The stochastic series avoids imposing that unsupported regularity.