Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 356 / 2 / a

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- Solution a

Solution

The two-dimensional Fokker-Planck equation is

\partial_{t} p = - a_{1} \partial_{x} p - \partial_{y} (a_{2} p) + \partial_{xx} p + \frac{1}{2} \partial_{yy} (σ^{2} p) = - \nabla \cdot J,

with Fokker-Planck probability current

J = (a_{1} p - \partial_{x} p, a_{2} p - \frac{1}{2} \partial_{y} (σ^{2} p)) .

The point initial condition is

p (x, y, 0) = δ (x) δ (y),

where

δ

is the Dirac delta function. The reflecting boundary condition for a diffusion is

J \cdot n = 0 on \partial Ω,

with

n

the outward unit normal. This zero-flux condition conserves the integral of the probability density function over the square.

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