The result from stochastic optimal control that we use in the paper ("verification theorem") originates mainly from the work of M. Pavon (1989) and P. Dai Pra (1991).
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.
julbern OP t1_ivyy0g1 wrote
Reply to comment by Benlus in [R] An optimal control perspective on diffusion-based generative modeling by julbern
The result from stochastic optimal control that we use in the paper ("verification theorem") originates mainly from the work of M. Pavon (1989) and P. Dai Pra (1991).
Perhaps it is best to start with the lecture notes of R. Van Handel (2007).
For books on the topic, I can further suggest:
Some more recent works in this direction are the following: