Quantitative Asymptotics for Time-Inhomogeneous Lévy-Driven SDEs with Asymptotically Vanishing Drifts
Abstract
In this work, we are concerned with a class of multi-dimensional time-inhomogeneous stochastic differential equations (SDEs) on Rd driven by pure-jump Lévy processes, where the drift coefficient b(t,x) satisfies limt→ ∞b(t,x) =0 for every x∈ Rd. On account of three regimes associated with the index α of large jumps corresponding to the driven Lévy noise, we investigate the quantitative asymptotics of the corresponding rescaled processes. More precisely, for α∈ (0,2), we prove that the rescaled processes, governed by time-inhomogeneous SDEs subject to additive processes, converge with respect to suitably chosen Wasserstein distances to time-homogeneous SDEs driven by symmetric α-stable processes. Notably, the driven noise in the limiting SDEs depends only on large jumps of the underlying additive processes. In case of α≥2, a phase transition occurs and a diffusive phenomenon arises. In particular, in the setting α>2, we establish the ergodicity of the rescaled process by means of asymptotic pseudotrajectories. The resulting time-homogeneous limiting SDEs are driven by Brownian motions, even though the transformed time-inhomogeneous SDEs are driven by (discontinuous) additive processes, and the effective noise intensity is determined by the entire Lévy measure of the original pure-jump process. As far as the critical case $α=2 $ is concerned, we demonstrate that the noise intensity of the limiting SDEs driven by Brownian motions relies merely on the large-jump part of the Lévy measure.
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BibTeX
@misc{bao2026quantitative,
title = {{Quantitative Asymptotics for Time-Inhomogeneous Lévy-Driven SDEs with Asymptotically Vanishing Drifts}},
author = {Jianhai Bao and Jian Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11657},
eprint = {2609.11657},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11657},
}
RIS
TY - UNPB
TI - Quantitative Asymptotics for Time-Inhomogeneous Lévy-Driven SDEs with Asymptotically Vanishing Drifts
AU - Jianhai Bao
AU - Jian Wang
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11657
UR - https://arxiv.org/abs/2609.11657
AB - In this work, we are concerned with a class of multi-dimensional time-inhomogeneous stochastic differential equations (SDEs) on R^d driven by pure-jump Lévy processes, where the drift coefficient b(t,x) satisfies lim_t→ ∞b(t,x) =0 for every x∈ R^d. On account of three regimes associated with the index α of large jumps corresponding to the driven Lévy noise, we investigate the quantitative asymptotics of the corresponding rescaled processes. More precisely, for α∈ (0,2), we prove that the rescaled processes, governed by time-inhomogeneous SDEs subject to additive processes, converge with respect to suitably chosen Wasserstein distances to time-homogeneous SDEs driven by symmetric α-stable processes. Notably, the driven noise in the limiting SDEs depends only on large jumps of the underlying additive processes. In case of α≥2, a phase transition occurs and a diffusive phenomenon arises. In particular, in the setting α>2, we establish the ergodicity of the rescaled process by means of asymptotic pseudotrajectories. The resulting time-homogeneous limiting SDEs are driven by Brownian motions, even though the transformed time-inhomogeneous SDEs are driven by (discontinuous) additive processes, and the effective noise intensity is determined by the entire Lévy measure of the original pure-jump process. As far as the critical case $α=2 $ is concerned, we demonstrate that the noise intensity of the limiting SDEs driven by Brownian motions relies merely on the large-jump part of the Lévy measure.
ER -