 Solution

ID: past-exam-of-the-flows-fluctuations-and-complexity-course-of-the-university-of-oxford/2016/4/b/solution

Write the source's noise terms consistently as

η_{R}, η_{P}

. Subtract the steady mean to get

\dot{δ R} = - γ_{R} δ R + η_{R}

. Use the Fourier convention

f (t) = \int d ω e^{- iω t} f (ω) / (2 π)

. After initial transients decay,

δ R (ω) = \frac{η _{R} ( ω )}{γ _{R} - iω} .

Fourier transforming the stationary white-noise covariance gives

⟨ η_{R} (ω) η_{R} (ω^{'})⟩ = 2 π q_{R} δ (ω + ω^{'}) .

Therefore the RNA fluctuation power spectral density is

S_{R} (ω) = q_{R} / (γ_{R}^{2} + ω^{2})

⟨ δ R^{2} ⟩ = \int_{- \infty}^{\infty} \frac{d ω}{2 π} \frac{q _{R}}{γ _{R}^{2} + ω ^{2}} = \frac{q _{R}}{2 γ _{R}} .

The positivity of

γ_{R}

ensures a stationary limit. The same calculation gives the autocovariance

⟨ δ R (t) δ R (0)⟩ = [q_{R} / (2 γ_{R})] e^{- γ_{R} ∣ t ∣}

: the decay rate filters the white forcing into a finite correlation time. Here the covariance convention is

q_{R} δ (t - t^{'})

; replacing it by

2 D_{R} δ (t - t^{'})

would identify

q_{R} = 2 D_{R}

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