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

The two-dimensional Fokker-Planck equation iswith Fokker-Planck probability currentThe point initial condition iswhere $\delta$ is the Dirac delta function. The reflecting boundary condition for a diffusion iswith $\mathbf{n}$ the outward unit normal. This zero-flux condition conserves the integral of the probability density function over the square.

Let ${\xi }_{1,n},{\xi }_{2,n}$ be independent random variables with the standard normal distribution. The unconstrained Milstein method step isThe $X$ noise is additive, so its Milstein correction vanishes. The diagonal, independent noise fields commute, so no cross iterated stochastic integral is required.

Implement each reflecting boundary condition for a diffusion by folding the proposed coordinate back into $[-1,1]$. One formula that also handles multiple overshoots isThe complete update isUnder the usual smoothness assumptions, the Milstein discretization has strong order of convergence one and weak order of convergence one. The reflection enforces the boundary pathwise.

The adsorption mechanism is uniform along the two vertical sides, and $X$ evolves independently of $Y$. Consequently the mean first-passage time depends only on the initial $x$-coordinate. Its Kolmogorov backward equation for $a_1=1$ is

The forward partially absorbing boundary condition for a diffusion is $\mathbf{J}\cdot \mathbf{n}=\kappa p$. The boundary term in the adjoint relation isbecause the $X$ diffusion coefficient is one. It vanishes for every admissible $p$ precisely whenFor $\kappa =1$, the two Robin boundary conditions are therefore

The general solution of the ordinary differential equation isThe right condition gives $C=2$, and the left gives $D=-2/e$. At the prescribed initial position,

With an absorbing boundary condition for a diffusion on each vertical side, the splitting probability $u(x)$ of hitting the left side before the right is harmonic for the Kolmogorov backward equation:For $a_1\ne 0$,The particle starts at $x=0$, soThe continuous limit at zero is $g(0)=1/2$, as required by reflection symmetry. A large positive drift drives the particle toward the right and gives $g(a_1)\to 0$; a large negative drift drives it toward the left and gives $g(a_1)\to 1$.