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A linearly stable $f_0$ has no zeros of the causal dielectric function in $\mathrm{Re}p>0$. Hence the inverse Laplace transform contours can approach $\sigma =0^{+}$, retaining the causal boundary value at $p=-i\omega +0^{+}$. To obtain a stationary forced response, further assume that the excited collective perturbations actually damp. This is the decay condition in the question's long-time instruction.

A transparent evaluation of the double integral uses the stationary power spectral density. The white noise covariance $2A_k\delta (t-t^{\prime })$ has temporal spectrum $2A_k$. Linear filtering by $1/ϵ$ therefore givesThe causal solution for $\delta f_k$ in a gives, after the initial transients have decayed,The real part can be taken because the full $k$ sum pairs conjugates. Using the distribution theory identity $\mathrm{Re}{\int }_0^{\infty }{e}^{ia\tau }d\tau =\pi \delta (a)$, with a positive vanishing convergence factor, samples the temporal spectrum at $\omega =kv$:The factor $1/2$ from the one-sided time integral cancels the $2$ in the noise covariance. The result is nonnegative, as a stationary velocity-space diffusion coefficient should be.

Qualification of stability. Absence of growing modes alone does not guarantee this stationary limit. For example, the electron waterbag distribution $f_0(v)=n_e{1}_{|v|<V}/(2V)$ hasas undamped collective roots. White noise drives their field variance to grow instead of settling. Thus the formula requires decay of excited modes, not merely absence of exponentially growing ones. For weak Landau damping, the waiting time can be much longer than ${\omega }_{pe}^{-1}$, and must still be short compared with evolution of $f_0$ for the frozen-background quasilinear approximation to apply.

Stochastic heating increases random particle kinetic energy through decorrelating electromagnetic kicks. A simple one-dimensional model makes the normalization explicit. Let $f(v,t)$ be a velocity distribution with number density $n=\int fdv$, and suppose constant-coefficient velocity-space diffusion givesThe coefficient has dimensions of squared velocity per time, and a freely diffusing component has variance $2Dt$. For sufficiently decaying tails, two applications of integration by parts yieldThe first velocity moment is unchanged, so this is growth of random motion rather than acceleration of the mean flow. In three independent isotropic components with coefficient $D$ in each, the heating rate per volume is instead $3mnD$. More generally each diagonal component contributes its own diffusion coefficient.

This idealization describes externally supplied, rapidly decorrelating forcing; its energy budget is not that of an isolated particle population. Screening, resonant self-consistent fields, collisions and velocity-dependent diffusion can change the result. In other plasma settings, stochastic heating also describes irregular orbits driven by spatial structure or overlapping resonances; that mechanism must be specified separately.