The generative process in this paper is given by an ODE and the diffusivity coefficient in the corresponding Fokker-Planck equation is thus zero. In this case, the verification theorem basically reduces to the instantaneous change of variables formula (Chen et al., 2018).

On the other hand, the solution to the Poisson equation (with homogeneous Dirichlet boundary condition) considered in the paper also has a stochastic representation based on an SDE with a corresponding stopping time (leading to "walk-on-spheres" methods). It would be quite interesting to merge these viewpoints.