Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 356 / 1 / a / Solution

The backward generator of the drift--diffusion is $\mathcal{L}=\alpha {\partial }_y+D{\partial }_y^2$. The survival probability satisfies the Kolmogorov backward equationwithThe target is absorbing, while reflection gives the Neumann boundary condition at $L$.