Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 13C / a

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- Solution a

Solution

Define the Multivariate Fokker-Planck probability current by

J_{i} = A_{i} P - \frac{1}{2} \frac{\partial}{\partial x _{j}} (B_{ij} P) .

Then

- \frac{\partial J _{i}}{\partial x _{i}} = - \frac{\partial}{\partial x _{i}} (A_{i} P) + \frac{1}{2} \frac{\partial ^{2}}{\partial x _{i} \partial x _{j}} (B_{ij} P),

so the Fokker-Planck equation is exactly the continuity equation

\partial_{t} P + \nabla \cdot J = 0

.

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