= Solution
With periodic or sufficiently decaying <boundary conditions>, <integration by parts> gives
$$
\frac{\delta F}{\delta\phi}=a\phi-\kappa\nabla^2\phi.
$$
Assume $\kappa\ge0$ for a stable quadratic <free energy>; the diffusive smoothing discussed below requires $\kappa>0$. In a periodic box of volume $V$, use the unitary convention $\phi_{\mathbf q}=V^{-1/2}\int_V\phi(\mathbf r)e^{-i\mathbf q\cdot\mathbf r}\,d\mathbf r$. A real field has $\phi_{-\mathbf q}=\phi_{\mathbf q}^*$. Since the <Fourier transform of a derivative> converts $\nabla^2$ to $-q^2$, each <Fourier mode> obeys
$$
\boxed{\dot\phi_{\mathbf q}=-r(q)\phi_{\mathbf q}+\eta_{\mathbf q},\qquad r(q)=\Gamma(a+\kappa q^2)>0.}
$$
It is an <Ornstein-Uhlenbeck process>. Multiplication by the integrating factor $e^{r(q)t}$ gives
$$
\boxed{\phi_{\mathbf q}(t)=e^{-r(q)t}\phi_{\mathbf q}(0)+\int_0^t e^{-r(q)(t-s)}\eta_{\mathbf q}(s)\,ds.}
$$
The noise integral is understood as an <Itô integral> with a deterministic kernel; writing $\eta\,ds$ is shorthand for the corresponding Wiener increment. Direct differentiation in the mild stochastic sense checks both the equation and the initial value. The stated Kronecker-delta noise covariance corresponds to this finite-volume normalization. Unlike a <conserved order parameter>, the zero <wavenumber> mode decays at the nonzero rate $\Gamma a$.
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