Quantitative Asymptotics for Time-Inhomogeneous L\'evy-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 $\R^d$ driven by pure-jump L\'evy 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 $\alpha$ of large jumps corresponding to the driven L\'evy noise, we investigate the quantitative asymptotics of the corresponding rescaled processes. More precisely, for $\alpha\in (0,2)$, 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 $\alpha$-stable processes. Notably, the driven noise in the limiting SDEs depends only on large jumps of the underlying additive processes. In case of $\alpha\ge2$, a phase transition occurs and a diffusive phenomenon arises. In particular, in the setting $\alpha>2$, 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 L\'evy measure of the original pure-jump process. As far as the critical case $\alpha=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 L\'{e}vy measure.