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Initial equilibrium at $a_I$ givesThe initial field and later noise are independent, so their cross terms vanish. DefineWith the supplied Gaussian white noise covariance, the noise contribution isUsing the decay rate found in part (c), it follows thatThis is the covariance interpolation in a Gaussian Model A quench: each mode forgets its initial equilibrium with relaxation time $1/r(q)$ and approaches the final equilibrium variance. Larger-$q$ modes relax faster.

If the physical refractive index is $n=1+\mu W$, then $n^2-1=2\mu W+{\mu }^2{W}^2$. In the parabolic wave equation, the last term produces $ik{\mu }^2\mathbb{E}[W^{2}E]/2$ in the mean equation. For unit-variance, finite-correlation fluctuations, $\mathbb{E}[W^{2}E]=m+O(\mu )$ in a weak-fluctuation expansion, giving a coherent phase drift $ik{\mu }^{2}m/2$ to second order. This term is absent if the model deliberately linearizes $n^2-1$ to $2\mu W$. It is not obtained by squaring ideal Gaussian white noise, whose pointwise square is not the finite-variance random variable $W^2$.

For additive Gaussian white noise and a time-even order parameter, a midpoint discretization of the Onsager–Machlup functional evaluates drift derivatives at the midpoint configurations. Reversing a trajectory visits the same midpoints in reverse order, so the noise-to-path Jacobian determinant is invariant under reversal. It may depend on the trajectory, and must be distinguished from the constant normalization of the noise measure.

For nonconserved order-parameter dynamics with $\dot{p}=-\Gamma \mu [p]+f$, the factor at one time step, apart from a path-independent power of the step size, isHere the fields have first been restricted to a finite spatial grid, and ${\mathcal{H}}_n$ is the Hessian matrix of the free energy.

For mixed conserved and nonconserved order-parameter dynamics, the required noises for a joint configuration and flux trajectory arewhere $\mathsf{D},\mathsf{G}$ represent the divergence and gradient, and $\Delta =\mathsf{DG}$ represents the Laplacian. The Jacobian matrix isTaking its Schur complement gives the joint factorFor periodic boundary conditions, discretizing the two spatial operators compatibly gives $\mathsf{D}=-{\mathsf{G}}^{\mathsf{T}}$, so $\Gamma I-M\Delta$ is the positive relaxation operator when $\Gamma ,M>0$. Reversal leaves ${\mathcal{H}}_n$ unchanged and reverses the sign of the flux. The product of the joint factors is consequently the same for both histories, justifying its cancellation from the joint path probability of an order parameter and its flux. The functional chain rule used in the action ratio holds in the continuum limit of this midpoint convention.

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.