Set
$$
\boxed{D=\frac\kappa2}.
$$
The stated curl-free condition says that $C_j=(\kappa^{-1})_{ji}a_i$ is a gradient, $C=\nabla\Psi$. Define
$$
\boxed{V=-2\Psi},
\qquad
\partial_jV=-2(\kappa^{-1})_{ji}a_i.
$$
Then $a=-D\nabla V$, and the <Fokker-Planck probability current> becomes
$$
J=aP-D\nabla P
=\boxed{-PD\nabla(V+\log P)}.
$$
This is the <potential condition for a Fokker--Planck equation>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 2 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-353/2/a/i/solution