Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 69 / 3 / b / Solution

Differentiate the product of Dirac delta functions defining $\tilde{P}$, regarding $B$ and $\sigma$ as independent density arguments. The random continuity equation isFor $0<s<t$, the causal functional derivatives follow from the explicit strain solution and $\tilde{B}(t)=B_0\exp [{\int }_0^t\tilde{\sigma }(u)du]$:Consequently,In the Furutsu–Novikov formula, the covariance $\kappa \delta (t-s)$ samples the upper endpoint of the causal time integral. Symmetric regularization of the white noise gives half the mass there. The $B$ response vanishes as $s\uparrow t$, while the strain response tends to one, so $\mathbb{E}[f(t)\tilde{P}(t)]=-(\kappa /2){\partial }_{\sigma }P$. Averaging the continuity equation yields the Fokker-Planck equationThere is no direct diffusion term in $B$: its equation contains the time integral of colored noise, whereas the strain equation receives Gaussian white noise. The same result follows from the Itô diffusion with drift $(\sigma B,-\sigma /\tau )$ and diffusion vector $(0,\sqrt{\kappa })$.