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The relative-phase drift $\dot{\theta }=\omega -ϵ\sin \theta$ describes competing frequency mismatch and phase coupling. For $0<\omega <ϵ$, the equilibrium points are $\arcsin (\omega /ϵ)$ and $\pi -\arcsin (\omega /ϵ)$, stable and unstable respectively. They merge in a saddle-node bifurcation at $\omega =ϵ$; greater mismatch gives running phase dynamics. Adding Gaussian white noise gives overdamped Langevin dynamics in a tilted washboard potential.

An unpinned uniform height driven by Gaussian white noise is Brownian motion. Its variance grows as ${\sigma }^{2}t$ and it has no stationary distribution on the real height axis for $\sigma >0$. Indeed, a stationary characteristic function would obey $\hat{\mu }(k)=\hat{\mu }(k)e^{-{\sigma }^2{k}^{2}t/2}$, incompatible with continuity at zero. Quotienting out the uniform height can still leave stationary shape fluctuations.

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.

For a parabolic wave equation driven by longitudinal Gaussian white noise with transverse covariance kernel $B$, the mean envelope obeys ${\partial }_{x}m=i{\Delta }_{\perp }m/(2k_0)-\gamma m$. This closure requires independent increments in the longitudinal direction; statistical homogeneity and one-point Gaussian distributions alone are insufficient.

Continuous-time Gaussian white noise is a random distribution rather than an ordinary finite-variance function at every time. Integrating it against deterministic kernels gives Gaussian random variables with covariance $q\int f(s)g(s)ds$. For independent isotropic components, a three-dimensional dot covariance has amplitude $3q$. An ordinary rapidly decaying force covariance approximates this model only above its correlation time.

For the flux $W_{ij}=-M{\partial }_i{\mu }_j+N_{ij}$ of a vector order parameter, microscopic reversibility requires independent Gaussian white noise components with covarianceThe current changes sign under time reversal. The resulting path-probability ratio contributes $-(k_{B}T{)}^{-1}\int W_{ij}{\partial }_i{\mu }_j$ to its logarithm; integration by parts converts this to the free-energy loss from the conserved dynamics.

For quadratic free energy with positive $a,\kappa$ and kinetic coefficient $\Gamma$, a deterministic initial Dirac delta distribution evolves under nonconserved order-parameter dynamics to the meanTaking the expectation removes the centered Gaussian white noise; the resulting diffusion-reaction equation is solved by the heat kernel. The integrated mean is $A{e}^{-\Gamma at}$, so spreading does not conserve the initial excess. The connected Fourier mode covariance grows from zero to $k_{B}T/(a+\kappa q^2)$ as $1-e^{-2\Gamma (a+\kappa q^2)t}$ by the Itô isometry. A realization retains equilibrium fluctuations after its mean has decayed. Pointwise continuum white-noise quantities require a coarse-graining cutoff.

For a trace-free isotropic Gaussian white noise velocity gradient with covariance $⟨{\sigma }_{im}(t){\sigma }_{jn}(t^{\prime })⟩={\kappa }_2\delta (t-t^{\prime })[{\delta }_{ij}{\delta }_{mn}-({\delta }_{im}{\delta }_{jn}+{\delta }_{in}{\delta }_{jm})/4]$, interpret the multiplicative equation as the Stratonovich integral limit of smooth velocity fluctuations. The causal Furutsu–Novikov formula has endpoint weight one half. It gives $P_t=({\kappa }_2/2){\partial }_i[(B^2{\delta }_{ij}-B_i{B}_j/2){\partial }_{j}P]$ for the vector probability density function.

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.

The Hasselmann stochastic climate model represents slowly responding climate variables driven by fast weather fluctuations as well as external forcing. Klaus Hasselmann's 1976 paper separates fast and slow variables, retains fluctuations lost by simple averaging, and explains why stabilizing climate feedbacks are needed to prevent indefinitely growing variability. Its general framework is broader than a single-temperature equation.^[ref]

Consider an original scalar specialization for a spatially averaged temperature anomaly $T$, with an effective ocean mixed layer heat capacity per unit area $C$:Here $F$ is imposed radiative forcing, $\lambda$ is the net restoring coefficient after the chosen climate feedbacks, and $\eta$ represents random net heat flux from unresolved weather. The anomaly energy per unit area is $CT$, so its tendency equals the net input. The weather term represents exchange with omitted degrees of freedom; it does not assert that an isolated ocean creates energy. $F$ and $\eta$ have units $\mathrm{W}{\mathrm{m}}^{-2}$, and $\lambda$ has units $\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$.

First set $\eta =0$, start from $T(0)=0$, and switch to constant $F_0>0$. The integrating factor $e^{\lambda t/C}$ givesFor $C=2.0\times 10^8\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$, $\lambda =2.0\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ and $F_0=4.0\mathrm{W}{\mathrm{m}}^{-2}$, the relaxation time is $10^8\mathrm{s}$, about $3.17$ years using 365.25 days per year. At $t=\tau$, $T=2(1-e^{-1})\mathrm{K}\simeq 1.264\mathrm{K}$, with eventual warming $2\mathrm{K}$. Doubling $C$ doubles the response time but leaves this equilibrium unchanged. This separates thermal inertia from climate sensitivity.

Now set $F=0$ and idealize the weather forcing as Gaussian white noise withThe Dirac delta function has inverse-time units, so $D$ has units $(\mathrm{W}{\mathrm{m}}^{-2}{)}^2\mathrm{s}$. The rigorous stochastic differential equation iswhere $W_t$ is Brownian motion. This is an Ornstein-Uhlenbeck process. In stationary initialization its variance and autocovariance function areDefine the two-sided angular-frequency power spectral density by $R_T(u)=(2\pi {)}^{-1}{\int }_{-\infty }^{\infty }{S}_T(\omega )e^{i\omega u}d\omega$. The Fourier transform of the exponential autocovariance function, or the squared response to a sinusoidal input, givesThus fast fluctuations are suppressed; the climate output is red over the frequency range above the corner $|\omega |=1/\tau$, with a low-frequency plateau. This is a colored noise spectrum without a periodic driver. The corner is $f_c=1/(2\pi \tau )$ in cycles per second, not $1/\tau$. The continuous white input is only an approximation on times longer than weather's correlation time.

For the same $C,\lambda$ and $D=10^8(\mathrm{W}{\mathrm{m}}^{-2}{)}^2\mathrm{s}$, $v_{\infty }=0.25{\mathrm{K}}^2$ and the stationary standard deviation is $0.50\mathrm{K}$. From fixed $T(0)=0$, the variance grows as $v_{\infty }(1-e^{-2t/\tau })$, even when the forced expected value is predictable. Integrating $S_T/(2\pi )$ over all angular frequencies returns $D/(C\lambda )$, checking the normalization independently. If $\lambda =0$ the unforced variance instead grows as $2Dt/C^2$; if $\lambda <0$, deviations amplify. Neither case has the stationary spectrum above.

A red spectrum alone cannot establish attribution of recent climate change: internal variability, changing forcing and measurement effects must be distinguished. This scalar model supplies a calculable baseline for variability and persistence, not a universal spectrum for all climate timescales or a full attribution analysis.

For nonconserved order-parameter dynamics, the positive kinetic coefficient $\Gamma$ converts the functional derivative of the free energy into a relaxation rate through $\dot{p}=-\Gamma \delta F/\delta p$. Microscopic reversibility fixes the accompanying Gaussian white noise variance by the Model A fluctuation-dissipation relation.

Langevin dynamics describes a dynamical variable driven by deterministic forces and random noise. Underdamped Langevin dynamics retains inertia, whereas overdamped Langevin dynamics neglects it. At thermal equilibrium, the fluctuation-dissipation relation for a Langevin particle relates Gaussian white noise strength to drag and temperature.

A linear fluctuating interface smooths height gradients by diffusion and receives additive Gaussian white noise. With periodic or Neumann boundary conditions, its nonzero Fourier modes are independent Ornstein-Uhlenbeck processes if the noise is diagonal in that basis. The unconstrained uniform height is a Brownian zero mode of a fluctuating interface.

A longitudinal Brownian field ${\mathcal{B}}_x(z)$ is a centered Gaussian random field withwhere $B$ is a valid transverse covariance kernel. For each fixed $z$ it is a scaled Brownian motion, and increments over disjoint longitudinal intervals are independent. Such a field drives a Markov model of wave propagation in a random medium. Its formal longitudinal derivative is Gaussian white noise in $x$, with transverse correlations described by $B$; it is not in general a Brownian sheet.

A parabolic wave equation driven by longitudinal Gaussian white noise with transverse covariance kernel $C$ has the displayed closed mutual coherence equation. The noise is the derivative of a longitudinal Brownian field with transverse covariance; the physical smooth-noise limit uses the Stratonovich integral. Conversion to the Itô integral supplies the attenuation drift and the cross variation supplies the $C({\mathbf{s}}_1-{\mathbf{s}}_2)$ term. For unit plane-wave incidence and transverse statistical homogeneity, $\Gamma =\exp [-k^2{\mu }^{2}x(C(0)-C({\mathbf{s}}_1-{\mathbf{s}}_2))]$. General Gaussian colored media do not have this local closure.

Add the random thermal force to the deterministic optical trap relaxation time model. In one stationary coordinate of a harmonic optical trap, the overdamped Langevin equation isHere $\xi$ is Gaussian white noise understood through its integrals, $\delta$ is the Dirac delta function, and $\zeta$ is the linear friction coefficient. The covariance coefficient has units ${\mathrm{N}}^2\mathrm{s}$; multiplying by $\delta (t-t^{\prime })$, of units ${\mathrm{s}}^{-1}$, gives squared force. This fluctuation-dissipation relation for a Langevin particle makes the stationary variance ${\sigma }^2=k_{B}T/k$, consistent with the equipartition theorem. The solution is an Ornstein-Uhlenbeck process with autocovariance functionIndeed, at positive lag the noise has zero correlation with the initial position, so $\zeta C_x^{\prime }=-k{C}_x$; its initial value is ${\sigma }^2$. This supplies both a decay time and an amplitude, making a stronger calibration test than a position histogram alone.

Specify the Fourier transform convention before using a power spectral density. For frequency $f$ in hertz, define $S_x^{(2)}(f)={\int }_{-\infty }^{\infty }{C}_x(t)e^{-2\pi ift}dt$ and the one-sided density $S_x^{(1)}(f)=2S_x^{(2)}(f)$ for $f>0$. Integration of the exponential givesThe position power spectral density has units ${\mathrm{m}}^2/Hz$; its integral is a variance, not a standard deviation. At $f=f_c$ it is half its zero-frequency plateau. Angular frequency conventions require a corresponding $2\pi$ conversion in both axis and integration measure. If $x=\beta (V-V_0)$ with position sensitivity $\beta$ in $m/V$, then $S_V^{(1)}=S_x^{(1)}/{\beta }^2$ in ${\mathrm{V}}^2/Hz$ before independent detector noise is added.

For an original invented calibration example, choose $a=0.500\mu \mathrm{m}$, dynamic viscosity $\eta =1.00\times 10^{-3}\mathrm{Pa}\mathrm{s}$, $T=300\mathrm{K}$, $k=0.0200pN/nm=2.00\times 10^{-5}N/m$, and $\beta =100nm/V$. The unbounded no-slip Stokes drag law givesThe corresponding voltage standard deviation is $0.14391\mathrm{V}$, and the position power spectral density plateau is $0.390369\mathrm{n}{\mathrm{m}}^2/Hz$, or $3.90369\times 10^{-5}{\mathrm{V}}^2/Hz$ in the detector. Recovering both $k=k_{B}T/{\sigma }^2$ and $k=2\pi \zeta f_c$ is an internal model check; actual agreement also tests the assumed temperature, drag and detector response.

The generator uses the exact stationary update $x_{j+1}=\rho x_j+\sigma \sqrt{1-{\rho }^2}{Z}_j$, with $\rho =e^{-\Delta t/\tau }$ and independent standard normal random variables $Z_j$. For a demeaned block of $N$ samples and its unnormalized discrete Fourier transform $X_j$, the two-sided periodogram is $\Delta t|X_j{|}^2/N$; doubling positive bins except the zero and Nyquist endpoints gives a one-sided density. Summing those bins times $\Delta f=1/(N\Delta t)$ returns the block mean square by discrete Parseval identity. Independent detector noise, finite exposure and aliasing would alter a measured spectrum and must enter a fit before interpreting $f_c$.

Write $C_j={\sum }_{t=1}^T{y}_t\cos ({\omega }_{j}t)$ and $S_j={\sum }_{t=1}^T{y}_t\sin ({\omega }_{j}t)$. The discrete Fourier transform coefficient is $C_j-i{S}_j$. Alternatively, by the given orthogonality relations, the harmonic obtained by linear least-squares projection has coefficients $a_j=2C_j/T$ and $b_j=2S_j/T$. Thus the source's periodogram is the squared amplitude of a fitted harmonic, scaled as power per unit angular frequency:It measures how much of the centered record varies at frequency ${\omega }_j$. For odd $T$, Fourier Parseval identity givesThe positive-frequency spacing is $2\pi /T$, so summing the power ordinates times this spacing estimates the total variance. The factor $1/(\pi T)$ corresponds to the one-sided spectral density on $(0,\pi )$; the usual two-sided periodogram is half as large.

If the observations are Gaussian white noise with common variance ${\sigma }^2$, the $2m$ harmonic sums are jointly normally distributed. Their means are zero even with a nonzero constant series mean, because each sine and cosine sums to zero. Orthogonality givesUncorrelated coordinates of a multivariate normal distribution are independent. Therefore all $C_j/\sqrt{T{\sigma }^2/2}$ and $S_j/\sqrt{T{\sigma }^2/2}$ are independent standard normals, andSince the chi-squared distribution with two degrees of freedom has mean two and variance four,Here $f_{+}$ is the one-sided white-noise spectral density. Its exact unbiasedness is accompanied by a nonconcentrating distribution: along Fourier frequencies tending to any interior frequency, $I/f_{+}$ has the same nondegenerate exponential law at every $T$. It therefore does not converge in probability to one, proving failure of statistical consistency, not just nonvanishing variance.

For a general stationary process, exact finite-sample unbiasedness is not implied. Expanding the Fourier square instead giveswhere $\gamma$ is the autocovariance function. Under suitable short-memory regularity this tends to $f_{+}(\omega )$ at frequencies ${\omega }_j\to \omega$. The periodogram is thus generally asymptotically unbiased, while the exact assertion above is for white noise.

Use local periodogram smoothing to reduce variance: average $L_T$ neighboring ordinates with weights summing to one, where $L_T\to \infty$ but $L_T/T\to 0$. At an interior frequency the shrinking frequency window makes the bias tend to zero for smooth $f_{+}$, while averaging many weakly dependent ordinates makes the variance of order $1/L_T$. For white noise with equal weights, the ordinates are exactly independent and the variance is exactly $f_{+}^2/L_T$. For a general process, impose suitable short-memory and fourth-order dependence assumptions, such as a stationary Gaussian model with sufficiently summable autocovariances. These conditions, together with smoothness, give a consistent smoothed estimator. Boundary frequencies require an appropriately one-sided or reflected window.

Smooth spectral density alone is not enough. For example let $y_t=A{ϵ}_t$, where $A>0$ is a nondegenerate random variable constant throughout the record, independent of independent unit-variance Gaussian ${ϵ}_t$, with $E[A^2]<\infty$. The unconditional autocovariance is zero off lag zero and its one-sided spectral density is the smooth constant $E[A^2]/\pi$. Nevertheless a long local average estimates the realized $A^2/\pi$, not that deterministic unconditional value. This nonergodic example explains why a dependence/ergodicity condition accompanies the smoothing argument.

For the turning point test of randomness, the null is independent identically distributed continuous observations. Let $J_t$ be one if $y_t$ is a strict local maximum or minimum among $y_{t-1},y_t,y_{t+1}$, for $2\le t\le T-1$, and set $P_T={\sum }_t{J}_t$. Three ranks are equally likely; the middle observation is an extremum in four of the six orders, giving $E[J_t]=2/3$ and $\mathrm{Var}(J_t)=2/9$. ThereforeIndicators more than two positions apart depend on disjoint observations and are independent. For neighboring indicators, both are one precisely when four observations alternate up-down-up or down-up-down. By the probability-integral transform, use four independent uniforms. The first pattern has probabilityhere $b=y_t$, $c=y_{t+1}<b$, with the two outer inequalities having probabilities $b$ and $1-c$. The reverse pattern has the same probability, so $E[J_t{J}_{t+1}]=5/12$ and $\mathrm{Cov}(J_t,J_{t+1})=5/12-4/9=-1/36$.

At separation two, the triples share only one endpoint value $c$. Conditional on it, their other observations are independent. The probability that either triple's central observation is a turning point isIt follows that $E[J_t{J}_{t+2}]={\int }_0^{1}r(c{)}^{2}dc=9/20$, so this covariance is $9/20-4/9=1/180$. Combining all terms for $T\ge 4$ givesThe standardized count is approximately standard normal for large $T$, by a central limit theorem for bounded 2-dependent indicators. A two-sided randomness test rejects for large absolute values ofToo few turns suggests persistent movement, while too many suggests alternation. Exact rank-permutation calibration is possible at small $T$. Ties invalidate the continuous-rank probabilities unless their handling is specified. The test probes this feature of randomness; nonrejection does not establish independence.

Put $\tau =m/\zeta$. The causal Green function of the displacement equation is $G(t)={\zeta }^{-1}(1-e^{-t/\tau })$ for $t>0$ and zero for $t<0$. Its derivative has the jump needed to give $m{G}^{\prime \prime }+\zeta G^{\prime }=\delta (t)$. For the specified zero position and velocity,For Gaussian white noise these are stochastic convolutions; their first two moments follow from the stated covariance even without assuming Gaussian higher moments. In particular $⟨x⟩=⟨\dot{x}⟩=0$, andThe required equipartition theorem limit therefore fixes the fluctuation-dissipation relation for a Langevin particle:Here $k_B$ denotes the printed Boltzmann constant $k$. Set $D=k_{B}T/\zeta$, the Stokes–Einstein relation. Squaring the displacement kernel givesConsequently ${\lim }_{t\to \infty }⟨x⟩=0$ and ${\lim }_{t\to \infty }⟨x^2⟩=+\infty$, with $⟨x^2⟩=2Dt-3D\tau +o(1)$ for positive temperature. There is no finite equilibrium variance of position in an unconfined fluid. The inertial Brownian displacement with zero initial velocity has short-time variance $2D{t}^3/(3{\tau }^2)+O(t^4)$, rather than the ballistic law associated with an initially thermalized velocity.

For constant $D$, the diffusion equation has fundamental density $P(x,t)=(4\pi Dt{)}^{-1/2}{e}^{-x^2/(4Dt)}$, whose mean is zero and variance is $2Dt$. It describes the large-time, overdamped limit of the Langevin dynamics. Under the usual Gaussian-force assumption, the finite-time position density is Gaussian with the variance above and obeys a time-dependent diffusion equation with coefficient $D(1-e^{-t/\tau }{)}^2$. It does not obey the constant-$D$ equation through the initial inertial transient. The given covariance alone determines the moments, not a Gaussian law; the density statement uses the standard Gaussian thermal-noise model.

Now write $\zeta (x)={\zeta }_0+\epsilon {\zeta }_0^{\prime }x$ and $x=x_0+\epsilon x_1+O({\epsilon }^2)$, with ${\tau }_0=m/{\zeta }_0$ and $D_0=k_{B}T/{\zeta }_0$. For a fixed realization of the additive force, the first-order equation isIf the force amplitude is instead adjusted to local thermal equilibrium, its first-order noise term has zero mean in the inertial Itô formulation and does not change the following first-order mean calculation. The mixed unperturbed moment isLet $M(t)=⟨x_1(t)⟩$. Averaging and solving the scalar forced relaxation equation, with $M(0)=M^{\prime }(0)=0$, givesThus the long-time first-order coefficient and local mean drift areLiterally $\lim M=-\infty$ for ${\zeta }_0^{\prime }>0$, $+\infty$ for ${\zeta }_0^{\prime }<0$, and zero for ${\zeta }_0^{\prime }=0$; the useful finite quantity is $\lim M^{\prime }$. Because the viscosity gradient is only a local expansion, the physical drift result applies in the intermediate window ${\tau }_0\ll t\ll L_{\zeta }^2/D_0$, where $L_{\zeta }={\zeta }_0/(\epsilon |{\zeta }_0^{\prime }|)$. Extrapolating the linear drag law to arbitrarily large displacements could make the drag negative and would invalidate the model.

The isothermal diffusion with position-dependent drag interpretation uses locally matched thermal noise and $D(x)=k_{B}T/\zeta (x)$. Integrating the printed Fickian diffusion equation against $x$, with vanishing boundary fluxes, givesSince $D^{\prime }(x)=-\epsilon k_{B}T{\zeta }_0^{\prime }/{\zeta }_0^2+O({\epsilon }^2)$, this reproduces the same drift after the inertial transient. Equivalently the overdamped Itô diffusion has drift $D^{\prime }$ and noise amplitude $\sqrt{2D}$. Omitting that drift would give $P_t=(DP{)}_{xx}$, which is a different Fokker-Planck equation and not the stated Fickian equation.

Noise conventions cannot be mixed silently. If the force covariance is held at the spatially constant value $F_0=2{\zeta }_0{k}_{B}T$ while drag varies, the local overdamped diffusion coefficient is instead $\tilde{D}=F_0/(2{\zeta }^2)$ and its Itô drift is ${\tilde{D}}^{\prime }/2$. Its density equation is $P_t={\partial }_x[\tilde{D}{P}_x+({\tilde{D}}^{\prime }/2)P]$, not the isothermal Fickian equation. This still gives the same displayed first-order mean drift at $x=0$, but has different spatial diffusion and equilibrium properties. The stated Fickian interpretation therefore requires local thermal-noise matching.

Physically, thermally driven excursions travel farther on the lower-drag side. Their displacement is correlated with velocity, so averaging the position-dependent friction leaves a net drift toward greater diffusivity, without an applied force. This is not a violation of equilibrium: in the isothermal Fickian model the drift and spatially varying noise cancel in the probability flux for a uniform density. It is a local diffusion-induced shift of the centroid of an initially localized particle distribution.

Interpret the multiplicative Gaussian white noise as the limit of a smooth short-correlation-time velocity field, hence in the Stratonovich integral convention. Put ${\partial }_i=\partial /\partial B_i$. For a realization, differentiating the field-space Dirac delta gives its field-space continuity equationThe velocity-gradient covariance has zero trace variance: contracting $T_{mn}^{ij}$ with $m=i,n=j$ gives $3-(9+3)/4=0$. Thus the gradient is trace free, consistently with the incompressible ideal induction equation.

A noise impulse in component ${\sigma }_{jn}$ changes ${\tilde{B}}_j$ by ${\tilde{B}}_n$ times that impulse. Its instantaneous retarded action on the Dirac delta is thereforeOnly earlier times affect the present field. In the white-noise limit the integral over this causal interval has ${\int }_0^t\delta (t-t^{\prime })d{t}^{\prime }=1/2$, corresponding to the symmetric short-time regularization. Applying the Furutsu–Novikov formula and then averaging givesSubstitute into the averaged field-space continuity equation:The differentiated $B_n$ term vanishes because ${\sum }_j{T}_{mj}^{ij}=0$, and the remaining contraction isConsequently the closed vector Fokker-Planck equation for Gaussian white-noise magnetic stretching isThis is a positive diffusion operator: the displayed matrix has eigenvalues $B^2$ on the two transverse directions and $B^2/2$ along $\mathbf{B}$. Replacing the endpoint half-weight by one would double the diffusion and fail to reproduce the later rate.

The strain is an Ornstein-Uhlenbeck process. Multiplying its equation by the integrating factor $e^{t/\tau }$ and using the zero initial condition givesThe second expression is a stochastic integral against Brownian motion. A deterministic linear functional of Gaussian white noise is Gaussian, and its expected value is zero. For $t\ge t^{\prime }$, the noise covariance givesFor $t^{\prime }\gg \tau$, the initial-condition term is negligible. The stationary autocorrelation is thereforeIts normalized exponential decay has correlation time $\tau$. The zero-start Ornstein-Uhlenbeck covariance also shows how stationarity is approached.