Tempered Langevin diffusions with polynomial-tailed targets: explosions, moment thresholds and ergodicity

Abstract

For tempered Langevin diffusions (TLDs) targeting polynomially tailed measures, the convergence rate is known to change from polynomial to exponential to uniform as the temperature rises through a critical value. We explain this transition by the tails of the marginal laws of the norm of the TLD, up to a second, larger critical temperature beyond which the TLD explodes. As the temperature increases, these tails change from stretched-exponential, through super-polynomial at the first critical temperature, to polynomial, mirroring the transition in the convergence rate. Moreover, this tail analysis gives a complete quantification of the convergence rates throughout the non-explosive regime, in both $f$-variation and Wasserstein distances.