发表机构
Center for Applied Mathematics, Tianjin University; School of Mathematics and Statistics & Key Laboratory of Analytical Mathematics and Applications (Ministry of Education) & Fujian Provincial Key Laboratory of Statistics and Artificial Intelligence, Fujian Normal University（天津大学应用数学中心; 福建师范大学数学与统计学院与分析数学及应用教育部重点实验室及福建省统计与人工智能省级重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究漂移渐近消失的时间非齐次Lévy驱动SDEs，按大跳指数α分三种情形，证明重标度过程在Wasserstein距离下收敛到由对称α-稳定过程或布朗运动驱动的时间齐次极限，并揭示相变与噪声强度依赖关系。
AI 中文摘要
本文研究了一类由纯跳Lévy过程驱动的定义在$\R^d$上的多维时间非齐次随机微分方程（SDEs），其中漂移系数$b(t,x)$满足对每个$x\in \R^d$有$\lim_{t\to \infty}b(t,x) =0$。基于驱动Lévy噪声对应的大跳指数$\alpha$的三种情形，我们研究了相应重标度过程的定量渐近性。更精确地，对于$\alpha\in (0,2)$，我们证明了由受加性过程支配的时间非齐次SDEs控制的重标度过程，在适当选取的Wasserstein距离意义下收敛到由对称$\alpha$-稳定过程驱动的时间齐次SDEs。值得注意的是，极限SDEs中的驱动噪声仅依赖于底层加性过程的大跳。在$\alpha\ge2$的情形下，发生相变并出现扩散现象。特别地，在$\alpha>2$的设置中，我们通过渐近伪轨迹建立了重标度过程的遍历性。尽管变换后的时间非齐次SDEs由（不连续的）加性过程驱动，但所得的时间齐次极限SDEs由布朗运动驱动，且有效噪声强度由原始纯跳过程的整个Lévy测度决定。至于临界情形$\alpha=2$，我们证明了由布朗运动驱动的极限SDEs的噪声强度仅依赖于Lévy测度的大跳部分。
英文摘要
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 $\lim_{t\to \infty}b(t,x) =0$ for every $x\in \R^d$. 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 $α\in (0,2)$, 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 $α\ge2$, 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.
Comments55 pages
