Machine-Learned Sampling of Conditioned Path Measures

We propose algorithms for sampling from posterior path measures $P(C([0,T],{\mathbb{R}}^d))$ under a general prior process. This leverages ideas from (1) controlled equilibrium dynamics, which gradually transport between two path measures, and (2) optimization in $\mathrm{\infty }$-dimensional probability space endowed with a Wasserstein metric, which can be used to evolve a density curve under the specified likelihood. The resulting algorithms are theoretically grounded and can be integrated seamlessly with neural networks for learning the target trajectory ensembles, without access to data.