  Recent advancements in diffusion models and diffusion bridges primarily focus
on finite-dimensional spaces, yet many real-world problems necessitate
operations in infinite-dimensional function spaces for more natural and
interpretable formulations. In this paper, we present a theory of stochastic
optimal control (SOC) tailored to infinite-dimensional spaces, aiming to extend
diffusion-based algorithms to function spaces. Specifically, we demonstrate how
Doob's $h$-transform, the fundamental tool for constructing diffusion bridges,
can be derived from the SOC perspective and expanded to infinite dimensions.
This expansion presents a challenge, as infinite-dimensional spaces typically
lack closed-form densities. Leveraging our theory, we establish that solving
the optimal control problem with a specific objective function choice is
equivalent to learning diffusion-based generative models. We propose two
applications: (1) learning bridges between two infinite-dimensional
distributions and (2) generative models for sampling from an
infinite-dimensional distribution. Our approach proves effective for diverse
problems involving continuous function space representations, such as
resolution-free images, time-series data, and probability density functions.
