Gaussian process

A Gaussian process is a stochastic process whose every finite vector of values has a multivariate normal distribution. Its law is determined by its mean and covariance functions.

Gaussian process regression estimates an unknown function by assigning its values a Gaussian process law, then conditioning on observed values, often with independent Gaussian noise. The covariance function expresses the proposed relation between nearby inputs; the conditional multivariate normal distribution gives the resulting Gaussian process regression posterior. Its conditional mean provides a prediction and its conditional variance quantifies uncertainty under the chosen model. The index set can represent spatial locations or other inputs as well as time.

Let $X_0=0$ and let $X$ be a centered Gaussian process whose increments are independent of the past filtration, with deterministic variance $v(t)$. Its displayed exponential is a positive martingale. Indeed, $X_t-X_s$ is a centered Gaussian random variable with variance $v(t)-v(s)$, so the Gaussian moment-generating function givesThe sign-reversed process has the same variance and the same conclusion. Continuity of paths is unnecessary for this conditional-expectation argument.

A centered continuous stationary scalar Gaussian process of positive variance can be Markov only if its covariance satisfies the displayed equation for $s,t\ge 0$. Gaussian conditioning makes the future conditional mean $C(t)X_s/C(0)$; the Markov property then forces the equation by conditioning a future observation on the present and past. Conversely this equation makes the Gaussian residual future independent of the past, proving the Markov property. In the nondegenerate decaying case the covariance is a single exponential. A sum $c_1{e}^{-a_1|t|}+c_2{e}^{-a_2|t|}$ with positive coefficients and distinct positive decay rates fails the equation and hence is not scalar Markov.

For independent standard normal variables ${\xi }_i$ and ${\sum }_i{k}_i(t{)}^2<\infty$ at each fixed time, the series defines $X_t$ in $L^2$. Every finite linear combination is an $L^2$ limit of Gaussian variables and is Gaussian. Thus this constructs a Gaussian process with the Gram covariance kernel $K$. It supplies no sample-path continuity without additional assumptions.

The canonical distance is the L2 norm of an increment of a Gaussian process. It is a pseudometric, since distinct parameters may represent equal random variables almost surely. Its metric covering number appears in the Dudley entropy integral. For standard fractional Brownian motion, $d(s,t)=|s-t{|}^H$; a normalization with twice the covariance function multiplies this distance by $\sqrt{2}$.

A fractional Brownian motion with Hurst exponent $H\in (0,1)$ is a centered Gaussian process with the displayed standard covariance function and a continuous modification. Its increment variance is $|s-t{|}^{2H}$. Some sources omit the factor $1/2$, multiplying the process by $\sqrt{2}$; the corresponding increment variance is then $2|s-t{|}^{2H}$. Its canonical pseudometric of a Gaussian process is proportional to $|s-t{|}^H$, which gives finite expected value of its absolute supremum on compact intervals through the Dudley entropy integral.

For fractional Brownian motion, $H\in (0,1)$ controls the variance $|t-s{|}^{2H}$ of increments and the scaling of its finite-dimensional distributions. Larger $H$ means smaller increment variance on intervals shorter than one and greater sample-path Hölder continuity.

For centered multivariate normal distributions, if $\mathbb{E}(U_i-U_j{)}^2\le \mathbb{E}(V_i-V_j{)}^2$ for every pair, then the displayed comparison holds. For a quick proof, make $U,V$ independent and set $Z_r=\sqrt{1-r}U+\sqrt{r}V$. Apply Gaussian integration by parts to $F_{\tau }(z)={\tau }^{-1}\log {\sum }_i{e}^{\tau z_i}$. With softmax function weights $p_i$ and $D=\mathrm{Cov}(V)-\mathrm{Cov}(U)$, the derivative of $\mathbb{E}{F}_{\tau }(Z_r)$ is $\frac{\tau }{4}\mathbb{E}{\sum }_{i,j}{p}_i{p}_j(D_{ii}+D_{jj}-2D_{ij})\ge 0$. Since $\max z_i\le F_{\tau }(z)\le \max z_i+{\tau }^{-1}\log n$, let $\tau \to \infty$. Singular covariance matrices follow by adding identical small independent normal noise to both vectors and taking a limit.

For centered multivariate normal distributions represented by vectors $U,V$ with equal corresponding variances and $\mathbb{E}[U_i{U}_j]\ge \mathbb{E}[V_i{V}_j]$ for all distinct indices, $\mathbb{P}(U_i\le u_i\text{ for all }i)\ge \mathbb{P}(V_i\le u_i\text{ for all }i)$ for every threshold vector. In particular the supremum of $U$ is stochastically smaller than the supremum of $V$. Equal corresponding variances are essential to this formulation. The related Sudakov-Fernique inequality compares expected values of maxima using variances of increments and does not require equal pointwise variances.

A Gaussian process has a mean-square derivative at $t$ when its difference quotients converge in L2 space. The mean-square convergence need not give differentiable sample paths. For a centered stationary process with twice continuously differentiable covariance function $K(t-s)$, the difference quotients are Cauchy in L2 space, since their covariance is a rectangular average of $-K^{\prime \prime }$. Their limits form a Gaussian process with covariance function $\mathbb{E}[Y(s)Y(t)]=-K^{\prime \prime }(s-t)$.

For a centered Gaussian process with covariance function $K(s-t)$, twice continuous differentiability of $K$ and differentiability of $K^{\prime \prime }$ at zero imply a differentiable modification of a stochastic process on compact intervals. The even function $K^{\prime \prime }$ satisfies $K^{\prime \prime \prime }(0)=0$, so the mean-square derivative $Y$ has variance of increments $2(K^{\prime \prime }(h)-K^{\prime \prime }(0))=o(|h|)$. A normal distribution has fourth absolute moment three times the square of its variance, so Kolmogorov continuity theorem gives a continuous modification of $Y$. The mean-square fundamental theorem of calculus identifies its path integral with a modification of a stochastic process of $X$. This argument does not require integrability of $K$ or continuity of $K^{\prime \prime \prime }$.

If an L2 space-valued map $X$ has a continuous mean-square derivative $Y$, then its increments equal the Bochner integral of $Y$. To see this, for each partition interval $[t_j,t_{j+1}]$, continuous linear observations and the scalar mean-value bound give $\Vert X(t_{j+1})-X(t_j)-(t_{j+1}-t_j)Y(t_j){\Vert }_2\le (t_{j+1}-t_j){\sup }_{u\in [t_j,t_{j+1}]}\Vert Y(u)-Y(t_j){\Vert }_2$. Sum these errors. Uniform continuity of $Y$ makes the total error tend to zero as the mesh tends to zero, while the Riemann sums converge to the Bochner integral. This proves the fundamental theorem of calculus in L2 space. If a Gaussian process derivative also has a continuous modification, its anchored path integral $X(s)+{\int }_s^t\tilde{Y}(u)du$ therefore produces a differentiable modification of a stochastic process of the original Gaussian process.

Gaussian process classification links latent variables from a Gaussian process to class probabilities. A binary probit regression likelihood is ${\prod }_i\Phi ((2y_i-1)f_i)$ under conditional independence. The posterior distribution of the latent values is not a normal distribution.

This covariance function is periodic with period $P>0$. Map time to the circle $u(t)=(\cos (2\pi t/P),\sin (2\pi t/P))$ and pull back a Gaussian kernel; since $\Vert u(t)-u(t^{\prime }){\Vert }^2=4{\sin }^2(\pi (t-t^{\prime })/P)$, this proves positive semidefiniteness. A Gaussian process with this kernel has $\mathrm{Var}(f(t+P)-f(t))=0$ and therefore repeats at every fixed phase. Replacing the inner $\pi$ by $2\pi$ changes the actual period to $P/2$.

A zero-mean Gaussian process need not have zero average in each realization. For the periodic covariance function, the constant Fourier component iswhere $I_0$ is a modified Bessel function. Replacing $f(t)$ by $f(t)-P^{-1}{\int }_0^{P}f(u)du$ gives a Gaussian process with kernel $k_P(t,t^{\prime })-c_0$. This is a positive-semidefinite kernel because it is a covariance after a linear transformation, and each sample has zero period average. The real integral representation of the modified Bessel function I0 yields the displayed constant.

For Gaussian-process latent values with mean vector $m$ and covariance matrix $K$, observed with independent Gaussian noise covariance $E$, integrating out the latent values gives the marginal distribution $N(m,K+E)$. Its density as a function of the mean and kernel parameters is the Gaussian-process marginal likelihood.

A Lipschitz function of a Gaussian vector has sub-Gaussian deviations from its mean, with scale controlled by the covariance operator and the Lipschitz constant. In particular, if the covariance operator norm is at most one and $f$ is one-Lipschitz, then universal constants $c,C>0$ give

For a standard Gaussian vector $G$ and a smooth Lipschitz function with $\Vert \nabla f\Vert \le K$, represent its centered value as the terminal martingale $P_{1-t}f(B_t)-P_{1}f(0)$. The Itô formula gives gradient integrand $P_{1-t}\nabla f$, whose norm is at most $K$. Its terminal quadratic variation is therefore at most $K^2$. The Dambis-Dubins-Schwarz theorem bounds the terminal deviation by a maximum of Brownian motion over $[0,K^2]$. Symmetry, the Brownian reflection principle, and the Gaussian tail bound give the displayed inequality. The square on the Lipschitz constant is essential by scaling.

For independent and identically distributed random variables $X,Y$ with a centered multivariate normal distribution, a continuously differentiable function $f$ with bounded gradient, and a convex function $\Psi$ for which the expressions are integrable, the displayed inequality holds. First apply Jensen inequality conditionally to $f(X)-f(Y)$. For the Gaussian rotation of independent copies, the chain rule gives $f(X)-f(Y)={\int }_0^{\pi /2}⟨\nabla f(U_{\theta }),V_{\theta }⟩d\theta$. A second application of Jensen inequality to the uniform measure on $[0,\pi /2]$, followed by the Gaussian rotation of independent copies, proves the inequality. The useful choice $\Psi (v)=e^{\lambda |v|}$ yields a Gaussian concentration inequality through an exponential moment of an absolute standard normal variable.

For a centered Gaussian process $(X_t{)}_{t\in T}$ with canonical pseudometric $d(s,t{)}^2=\mathbb{E}|X_s-X_t{|}^2$,where $N(ϵ,T,d)$ is the covering number.

If a centered separable Gaussian process has finite expected supremum $m$ and ${\sigma }^2={\sup }_t\mathrm{Var}(X_t)$, then

An isonormal Gaussian process over a real Hilbert space $H$ is a centered Gaussian process $(W(h){)}_{h\in H}$ satisfyingIt turns the geometry of $H$ into the covariance structure of a Gaussian family.

For any real separable Hilbert space $H$, any orthonormal basis $(e_j)$ and independent standard normal random variables $(Z_j)$, the displayed series converges almost surely for each fixed $h\in H$ and in L2 space. Indeed, the Parseval identity for a Hilbertian basis gives ${\sum }_j⟨h,e_j{⟩}^2=\Vert h{\Vert }^2$, so the partial sums form an L2 space-bounded martingale. The martingale convergence theorem supplies both limits. Every finite linear combination has the normal distribution with variance $\Vert {\sum }_j{u}_j{h}_j{\Vert }^2$. Thus its finite-dimensional characteristic function is $\exp (-\Vert {\sum }_j{u}_j{h}_j{\Vert }^2/2)$, independent of the chosen orthonormal basis. This proves equality of the finite-dimensional distributions of the resulting isonormal Gaussian processes; pathwise equality needs a coupling.

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.

Gaussian white noise is a generalized Gaussian process with zero mean and covariance proportional to a Dirac delta function in time, and possibly in space. It is the formal time derivative of Brownian motion and has independent fluctuations on disjoint time intervals.

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.

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.

The Gaussian sequence model observeswhere the $g_k$ are independent standard normal variables and the unknown signal commonly lies in ${\ell }^2$. It is the coordinate representation of a Gaussian white-noise experiment.

A Bernstein-von Mises theorem says that a suitably centered and scaled posterior distribution converges to the normal distribution prescribed by the local likelihood. For a fixed linear functional in a Gaussian sequence model and a prior with locally flat positive density, the limiting posterior variance is the squared ${\ell }^2$ norm of the functional's coefficient vector.

A Gaussian net test covers a composite alternative by finitely many metric balls and takes the maximum of the likelihood-ratio tests against their centers. Gaussian tail bounds control each test, while a union bound costs the logarithm of the covering number.

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.

A centered continuous Gaussian process that is also a martingale has independent increments. Every future increment has zero covariance with every finite vector of past values by the martingale property; uncorrelated jointly normal variables are independent.

Every centered continuous Gaussian martingale starting at zero is a deterministic time change of Brownian motion. Its clock is the continuous increasing function $f(t)=\mathbb{E}[M_t^2]$, and $[M{]}_t=f(t)$.

If a continuous local martingale starts at zero and has deterministic continuous quadratic variation $[M{]}_t=f(t)$, the Dambis-Dubins-Schwarz theorem gives $M_t=B_{f(t)}$. It is therefore a centered Gaussian process. The converse follows from independent increments of a Gaussian martingale.

The stationary exponential covariance function isThe corresponding Gaussian process is Markov and is a stationary Ornstein-Uhlenbeck process.

A Gaussian random field is a spatially indexed Gaussian process; its law is determined by its mean and covariance kernel.

A centered Gaussian random field with the displayed covariance. It has independent increments over disjoint rectangles. At fixed positive $T$, $W(t,T)/\sqrt{T}$ is standard Brownian motion in $t$. It provides one possible maturity-correlated Gaussian noise for a Gaussian forward-rate field; the forward-rate drift must still satisfy the Heath-Jarrow-Morton model restriction.

Let $X$ be a standard Gaussian random variable and $S_j$ independent symmetric signs, independent of $X$. The stationary sequence $W_j=S_{j}X$ has standard Gaussian marginals and identity covariance, but is not jointly Gaussian. For two coordinates its moment-generating function is $(1+e^{2t^2})/2$, rather than $e^{t^2}$. Consequently a field's Gaussian one-point densities and two-point covariance alone do not justify an exponential moment of a Gaussian linear functional.

If $G$ is a centered Gaussian random field, the transformation $G+\alpha (G^2-⟨G^2⟩)$ retains zero mean and introduces a three-point connected correlation function at first order in $\alpha$. Wick contractions give $2\alpha$ times the sum of products of two covariances. The corresponding local cosmological model uses $\alpha =3f_{\mathrm{NL}}/5$.

If a linear functional $Z=\int a(\mathbf{r})V(\mathbf{r})d\mathbf{r}$ of a real Gaussian random field exists in mean square, it is a Gaussian random variable. Completing the square in its density givesFor a complex linear functional $\varphi$, apply this to $Z=2\mathrm{Re}\varphi$; then $\frac{1}{2}\mathrm{Var}Z=\mathbb{E}|\varphi -\mathbb{E}\varphi {|}^2+\mathrm{Re}\mathbb{E}(\varphi -\mathbb{E}\varphi {)}^2$.

For a real standard Gaussian random vector $Z$, a real symmetric matrix $Q$, and $I-2Q$ strictly positive,Diagonalize $Q$ and complete the square in each coordinate. Failure of strict positivity makes the positive exponential integral divergent. The same formula extends to a self-adjoint trace-class quadratic form of an isonormal Gaussian process, with square-summable linear coefficients and a Fredholm determinant. A linear-plus-quadratic Gaussian random field functional is reduced to this form by its covariance square root.

A stationary Gaussian random field has Gaussian finite-dimensional distributions invariant under common spatial translations. Its mean is constant and its two-point covariance depends only on the separation of the points.

For a zero-mean stationary Gaussian random field $V(\mathbf{r})$, the autocorrelation function iswhich is independent of $\mathbf{r}$. Its Fourier transform is the spatial power spectral density.

For $W=n^2-⟨n^2⟩$, the covariance function is $⟨n^2{n}^{\prime 2}⟩-⟨n^2⟩⟨n^{\prime 2}⟩$. It involves fourth moments and is not determined by the two-point correlation of $n$ in general. For a centered Gaussian pair the fourth-moment identity reduces it to $2⟨n{n}^{\prime }{⟩}^2$. This extra Gaussian hypothesis must be stated before using that reduction.

For a centered Gaussian random field $W$ with covariance $C$ and a sufficiently regular functional $F[W]$, the Furutsu--Novikov formula is the functional integration by parts identityIt closes moment equations for systems driven multiplicatively by Gaussian fluctuations.

When a random medium's longitudinal correlation length is much shorter than the scale on which a propagating envelope changes, the Markov approximation replaces the longitudinal covariance by white noise with the same integrated covariance. Field-moment equations then become local in propagation distance.

The Gaussian free field is a centered Gaussian random field whose covariance is the Green kernel of a Laplace operator. It has discrete versions on graphs and continuum versions as a random generalized function on a domain.

The discrete Gaussian free field on a transient weighted graph is the centered Gaussian random field whose covariance kernel is the random-walk Green function of a transient weighted graph.

A Gaussian free field with Dirichlet boundary condition is fixed to zero on a boundary set and has density proportional to the exponential of minus one half of its discrete Dirichlet energy on the remaining vertices.

Conditionally on all other values, a discrete Gaussian free field value is Gaussian with mean equal to the weighted average of its neighbors and variance equal to the reciprocal local precision.

Conditioning a Gaussian free field on its values along a boundary leaves an independent zero-boundary Gaussian free field plus the harmonic extension of those boundary values in each remaining component.

The continuum Gaussian free field on a planar domain $D$ is a centered Gaussian process indexed by finite-energy measures or test functions, with covariance given by the Green function of the Laplacian on $D$.

If $\varphi :D\to \tilde{D}$ is a conformal map, pulling back a zero-boundary Gaussian free field on $\tilde{D}$ gives a zero-boundary Gaussian free field on $D$. This follows from the conformal invariance of the planar Dirichlet Green function.

For finite Borel measures $\rho ,\nu$ of finite Green energy, the zero-boundary Gaussian free field $h$ on $D$ is characterized bywhere $G_D$ is the zero-Dirichlet Green function of the Laplacian.

For a Gaussian free field with boundary heights $-\lambda$ and $+\lambda$ on the two sides of the starting point, its distinguished zero-height interface has chordal ${\mathrm{SLE}}_4$ law. Conditional on an initial curve segment, the field is a zero-boundary field in the slit domain plus $m_t(z)=\lambda -(2\lambda /\pi )\arg (g_t(z)-W_t)$. Its mean bracket equals minus the variation of its conditional Green covariance. Subtracting the initial harmonic function gives the corresponding coupling to a zero-boundary field.

If a continuous mean martingale $M$ and a nonnegative finite-variation process $V$ satisfy $d⟨M⟩=-dV$, then the Itô formula shows $\exp (iM-V/2)$ is a bounded complex martingale. Its terminal expectation establishes an entire Gaussian distribution with the initial mean and variance. This is stronger than matching only the first two moments in a random-domain field construction.

A continuum field level line is specified by the harmonic boundary heights it creates and the conditional zero-boundary field remaining on each side. Since a Continuum Gaussian free field is a random distribution, this is not the pointwise set of its zeros. For the height jump appropriate to the Green-function normalization, the zero-height line between opposite Dirichlet heights is chordal ${\mathrm{SLE}}_4$.

For a test function $\varphi$ on $D$, let $f_{\varphi }=-2\pi {\Delta }^{-1}\varphi \in H_0^1(D)$. The pairingis a centered Gaussian random variable with variance

For an orthonormal basis $(e_n)$ of $H_0^1(D)$, the zero-Dirichlet Green function of the Laplacian is the weak kernel represented byConsequently, whenever the terms are defined and the right side is finite,

If a measure $\rho$ has finite Green energy, thenconverges in $L^2$ for any orthonormal basis $(e_n)$ of $H_0^1(D)$ and independent standard normal coefficients ${\xi }_n$. The limit is centered Gaussian with variance equal to the Green energy of $\rho$ and does not depend on the chosen basis.

For an open $U\subset D$, the field decomposes as $h=h_U+h^{D\setminus U}$, where $h_U$ is an independent zero-boundary Gaussian free field on $U$ and $h^{D\setminus U}$ is harmonic on $U$ and agrees with the outside field in the distributional sense.

A random set is local for a Gaussian free field if, conditionally on the set and its associated harmonic function, the remaining field is an independent zero-boundary field in its complement. This extends the deterministic Domain Markov property of the Gaussian free field to appropriate random revealed domains. Arbitrary field-dependent random sets need not have this property.

Let ${\rho }_t$ be uniform measure on the circle of radius $e^{-t}$ around the origin in the unit disc. The process $X_t=(h,{\rho }_t)$ is the circle-average process. The Domain Markov property of the Gaussian free field and the mean value property for harmonic functions imply that its increments are independent and stationary.

The circle-average process of a zero-boundary Gaussian free field in the unit disc has the law of a constant multiple of Brownian motion. It is a continuous centered Gaussian process with stationary increments and independent increments and starts at zero.

For a measure $\mu$ and Green kernel $G$, its Green energy isFinite Green energy is the variance condition needed to pair a Gaussian free field with $\mu$.

Pinning a Gaussian free field at a vertex $o$ gives the field with value zero at $o$ and covarianceIt is independent of the removed scalar Gaussian mode at $o$.

Every linear subspace that is measurable for the cylinder sigma-algebra has probability zero or one under a centered Gaussian process.

A Gaussian measure on a real topological vector space is a Borel probability measure whose image under every continuous linear functional is a one-dimensional normal distribution.

A Gaussian random variable in a Banach space taking values in a separable Banach space satisfies the displayed exponential-integrability bound for some positive $a$. In particular every finite power of its norm has a finite expected value. The value of $a$ depends on the Gaussian measure; the theorem does not assert this for every positive $a$.

For a centered Gaussian random variable in a Banach space, its first Gaussian chaos is the closed linear subspace of L2 space generated by its continuous linear observations. Every element is a centered Gaussian random variable, and every finite collection has a multivariate normal distribution: approximate in L2 space and pass to the characteristic function. The first Gaussian chaos supplies the scalar coordinates of its Cameron-Martin space of a Gaussian random variable in a Banach space.

A random variable $X$ in a real separable Banach space $B$ is Gaussian when $\ell (X)$ has a normal distribution for every continuous linear functional $\ell \in B^{\ast }$. It is centered when all these expected values vanish. Its probability law is a Gaussian measure. Degenerate normal distributions are allowed.

Suppose $B$ is a nonzero separable Banach space, the Cameron-Martin space of a Gaussian random variable in a Banach space is a dense subset of $B$, and every centered ball of positive radius has positive probability. Then $t\mapsto \mathbb{P}(\Vert X{\Vert }_B\le t)$ is strictly increasing on $(0,\infty )$. For $0<a<b$, choose $h$ in that Cameron-Martin space of a Gaussian random variable in a Banach space with $\Vert h{\Vert }_B=(a+b)/2$ and $0<\delta <(b-a)/2$. The closed ball of radius $\delta$ centered at $h$ lies in the annulus $a<\Vert x{\Vert }_B<b$ and has positive probability by the symmetric Gaussian translation lower bound. The nonzero assumption is necessary: on $B=\{0\}$ the distribution function equals one everywhere.

For $\mu =\mathcal{N}(0,\Sigma )$ with injective covariance operator of a Gaussian measure, its translated law ${\mu }_h=\mathcal{N}(h,\Sigma )$ is an equivalent probability measure to $\mu$ if and only if $h\in E_{\mu }$, the Cameron-Martin space of a Gaussian measure. For such $h$,The series has mean-square convergence and converges almost surely under $\mu$, with normal distribution $\mathcal{N}(0,\Vert h{\Vert }_{E_{\mu }}^2)$. Its exponential density has expected value one by the moment-generating function of a normal distribution. Finite-dimensional projections give the formula by ratios of multivariate normal densities; convergence of these likelihood ratios gives the infinite-dimensional result. The expression $⟨h,x{⟩}_{E_{\mu }}$ is only formal when the sample is outside $E_{\mu }$. For $h\notin E_{\mu }$, translation gives mutually singular measures.

For a centered Gaussian random variable in a Banach space, a Borel set $C=-C$, and $h$ in its Cameron-Martin space of a Gaussian random variable in a Banach space, the Cameron-Martin theorem for a Gaussian measure gives the translated density $e^{-\hat{h}(X)-\Vert h{\Vert }_H^2/2}$. The joint law of $(X,\hat{h}(X))$ is invariant under simultaneous negation. Averaging the two density formulas therefore replaces $e^{-\hat{h}(X)}$ by $\cosh \hat{h}(X)\ge 1$, proving the lower bound. No convexity of $C$ is required.

For a centered Gaussian measure on a real Hilbert space with injective covariance operator of a Gaussian measure $\Sigma$, its Cameron-Martin space is $E_{\mu }=\mathrm{Ran}{\Sigma }^{1/2}$, with inner product $⟨h,g{⟩}_{E_{\mu }}=⟨{\Sigma }^{-1/2}h,{\Sigma }^{-1/2}g⟩$. In an orthonormal basis satisfying $\Sigma e_j={\lambda }_j{e}_j$, it consists of exactly the vectors $h$ for which ${\sum }_j{h}_j^2/{\lambda }_j<\infty$. It is complete in this stronger norm even if its range is not closed in the ambient norm. Its importance is that it describes precisely the translations that preserve the null sets of the Gaussian measure, as in the Cameron-Martin theorem for a Gaussian measure. For generalized Gaussian white noise over $L^2$, the corresponding Cameron-Martin space is L2 space itself.

Let $\mathcal{G}$ be the first Gaussian chaos of a centered Gaussian random variable in a Banach space. By Fernique's theorem, $Sg=\mathbb{E}[Xg]$ is a Bochner integral in $B$. The map $S$ is injective, because $Sg=0$ implies $\mathbb{E}[g\ell (X)]=0$ for every continuous linear functional $\ell$, hence $g=0$. Its range becomes a Hilbert space under $⟨Sg,Sv{⟩}_H=\mathbb{E}[gv]$. The embedding in $B$ is continuous by Cauchy-Schwarz inequality. Its reproducing property is $⟨h,S\ell (X){⟩}_H=\ell (h)$. Thus this is the Reproducing kernel Hilbert space of the random variable, including degenerate laws. For $h=Sg$, the scalar coordinate $\hat{h}(X)=g$ has variance $\Vert h{\Vert }_H^2$; it is generally a measurable linear coordinate rather than a continuous functional on $B$.

The covariance operator $K$ of a Gaussian measure on a real Hilbert space satisfies $⟨Kh,g⟩=\mathrm{Cov}(⟨U,h⟩,⟨U,g⟩)$. A positive self-adjoint operator is the covariance of a Hilbert-space-valued Gaussian random variable exactly when it is trace class.

If $K{e}_j={\lambda }_j{e}_j$ with ${\lambda }_j\ge 0$ and ${\sum }_j{\lambda }_j<\infty$, thenconverges in mean square and almost surely in the Hilbert space, and has Gaussian law $N(m,K)$.

A Bayesian inverse problem combines a prior measure ${\mu }_0$ on an unknown $u$ with a likelihood for observed data $y$. When the likelihood is proportional to $e^{-\Phi (u;y)}$, Bayes' formula gives

Suppose an unknown has prior distribution $\Pi$ and its conditional observation law $P_u$ has jointly measurable Radon-Nikodym derivative $L(u,m)$ relative to a fixed observation law $P_0$. If $L>0$ for $\Pi (du)P_0(dm)$-almost every pair, thendefines the posterior distribution for almost every observation. Indeed, $\int L(u,m)P_0(dm)=1$ for almost every $u$, so Tonelli theorem gives $\int Z(m)P_0(dm)=1$. Hence $Z$ is finite almost everywhere, and positivity of $L$ and Fubini's theorem give $Z>0$ almost everywhere. The data law is $Z{P}_0$, and integration of the proposed posterior against this data law recovers the joint law. This proves the conditional distribution property and also shows that the data law is an equivalent probability measure to $P_0$.

A Bayesian inverse problem is well posed in total variation when a unique posterior exists for every datum and the map from data to posterior is continuous in total variation distance.

A Bayesian inverse problem is well posed in Hellinger distance when every datum determines a unique posterior and the posterior depends continuously on the data in Hellinger distance. Since posterior measures are dominated by the prior and squared Hellinger distance is at most twice total variation distance, total-variation well-posedness implies Hellinger well-posedness.

An Ornstein-Uhlenbeck process combines mean reversion with continuing random fluctuations. In the scalar model, its stochastic differential equation iswhere $W$ is a standard Brownian motion. The drift points towards $\mu$: it is negative above $\mu$ and positive below it. The coefficient $\theta$ sets the speed of mean reversion, while $\sigma$ sets the strength of the random forcing. We consider a fixed or normal initial value independent of future Brownian increments, giving a Gaussian process and a Markov process.

For a stipulated example, use $\theta =1$, $\mu =2$, $\sigma =\sqrt{2}$ and the fixed initial value $X_0=6$, in arbitrary consistent state and time units. Then the expected value relaxes as $2+4e^{-t}$ and the variance grows as $1-e^{-2t}$. At time one they are about $3.4715$ and $0.8647$; the limiting variance is one. The sample paths keep fluctuating even after the expected value is close to two. The Ornstein-Uhlenbeck transition distribution describes this evolving uncertainty; Ornstein-Uhlenbeck stationary initialization instead starts with that limiting normal distribution already in place.

A physical motivation is the velocity of a particle undergoing Brownian motion: linear drag removes a persistent initial velocity, while random forcing continually renews it. The original calculation gives a normal distribution for the velocity after subtracting its exponentially decaying initial contribution, with decay rate equal to the friction coefficient divided by the particle's mass.^[ref] A position coordinate in an overdamped harmonic optical trap gives another interpretation of the same scalar equation, with $\theta =k/\zeta$ for spring constant $k$ and drag coefficient $\zeta$. Its physical assumptions and connection to measurements are developed in thermal calibration of an optical trap.