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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.
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BibTeX
@misc{mijatovic2026tempered,
title = {{Tempered Langevin diffusions with polynomial-tailed targets: explosions, moment thresholds and ergodicity}},
author = {Aleksandar Mijatović and Haokai Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30024},
eprint = {2610.30024},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30024},
}
RIS
TY - UNPB
TI - Tempered Langevin diffusions with polynomial-tailed targets: explosions, moment thresholds and ergodicity
AU - Aleksandar Mijatović
AU - Haokai Xie
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30024
UR - https://arxiv.org/abs/2610.30024
AB - 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.
ER -