Abstract

We characterise the rate of convergence of three classes of ergodic diffusions with multiplicative noise: (1) Reflected processes in generalised parabolic domains; (2) tempered Langevin diffusions; (3) forward processes in denoising diffusions of generative modelling. In this talk we show that these disparate looking models share the same underlying structure, which determines their respective rates of convergence to stationarity. (1) Varying multiplicative noise in reflecting diffusions leads to wildly different convergence rates to stationarity. (2) Tempered Langevin diffusions with fixed invariant measure exhibit significant improvements in the rates of convergence to stationarity with the addition of unbounded multiplicative noise. (3) We describe denoising diffusion probabilistic models (DDPMs) representing a recent advancement in generative AI, used in platforms such as ChatGPT. DDP Ms are contingent on the convergence to stationarity of an underlying forward diffusion, initialised at a high dimensional multi-modal data distribution. We establish a cut-off phenomenon for the convergence of the Ornstein-Uhlenbeck forward process used in the practical applications of DDP Ms and compare it with a large class of diffusions with multiplicative noise. Our results prove that in this case, unlike (1) and (2), the OU process (with additive noise) is hard to beat.