 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-8/3/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

For the normalized standard Gaussian density,

\nabla γ = - v γ, Δ γ = (∣ v ∣^{2} - d) γ, \nabla \cdot (v γ) = (d - ∣ v ∣^{2}) γ .

The two terms cancel, so the Ornstein-Uhlenbeck Fokker-Planck equation has stationary density for a Fokker-Planck equation

\partial_{t} γ = 0, Δ γ + \nabla \cdot (v γ) = 0.

- \nabla γ - v γ

vanishes identically. The normalization follows from the Gaussian integral in each coordinate.

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