HJ-sampler: A Bayesian sampler for inverse problems of a stochastic process by leveraging Hamilton-Jacobi PDEs and score-based generative models

Understand

The interplay between stochastic processes and optimal control has been extensively explored in the literature.

• With the recent surge in the use of diffusion models, stochastic processes have increasingly been applied to sample generation.
• This paper builds on the log transform, known as the Cole-Hopf transform in Brownian motion contexts, and extends it within a more abstract framework that includes a linear operator.
• Within this framework, we found that the well-known relationship between the Cole-Hopf transform and optimal transport is a particular instance where the linear operator acts as the infinitesimal generator of a stochastic process.