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

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 $g(a_1)\to 1$.