Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 4 / c

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- Solution c

Solution

Let

τ = ∣ t - t^{'} ∣

. Integrating the equation for

x

and using independence of the two noises gives

R (τ) = C^{2} τ + \int_{0}^{τ} \int_{0}^{τ} f_{0}^{2} e^{- α ∣ s - s^{'} ∣} d s d s^{'} .

With the supplied integral,

R (τ) = C^{2} τ + \frac{2 f _{0}^{2}}{α ^{2}} (ατ - 1 + e^{- ατ}) .

For

ατ ≪ 1

, the active contribution is ballistic,

f_{0}^{2} τ^{2} + O (τ^{3})

, in addition to the Brownian term. For

ατ ≫ 1

,

R (τ) = (C^{2} + \frac{2 f _{0}^{2}}{α}) τ - \frac{2 f _{0}^{2}}{α ^{2}} + o (1),

so the long-time motion is diffusive with an enhanced diffusion coefficient.

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