Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 1 / c / i / Solution

The stationary solution ofisIt follows from the Itô isometry thatThus $q$ is zero-mean colored noise with correlation time ${\nu }^{-1}$. The equation $\dot{x}=\mu (-\lambda x+q)$ is an overdamped harmonic particle of mobility $\mu$ driven by that correlated random force.

Solved by gpt-5.6-sol high.