Sampling from probability distributions is a fundamental challenge in physics and data science. State-of-the-art methods often rely on building efficient dynamics for probability measures. This talk examines the analysis and design of such dynamics, drawing insights from and targeting applications in high-dimensional scientific computing.

In the first part, we uncover and analyze a novel 'delocalization of bias' phenomenon in MCMC with Langevin dynamics. While sampling bias increases with dimensionality in full coordinates, individual coordinates can exhibit nearly dimension-independent behavior. This finding suggests that the curse of dimensionality in sampling may be mitigated at the level of low-dimensional marginals. In the second part, we propose a generative diffusion dynamics design for probabilistic forecasting, focusing on benchmark applications in stochastically forced Navier-Stokes equations. We prove that a specific design of diffusion coefficients minimizes novel statistical errors at the level of path measures and yields Föllmer processes, which also offer a Bayesian interpretation of the optimal design. We conclude with a real-data scientific application in black hole imaging, where we combine generative diffusion dynamics with MCMC for rigorous posterior sampling. Overall, the talk demonstrates how mathematical understanding and methodological design of high-dimensional sampling dynamics can synergize with insights and applications in scientific computing.