Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 8 / 3 / a

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- Solution a

Solution

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.

Equivalently its Fokker-Planck probability current

- \nabla γ - v γ

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

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