| Abstract: |
| 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 $x\in \R^d$. Distinguishing three regimes according to the index $\alpha$ of large jumps for the driven L\`evy noise, we investigate the quantitative asymptotic behavior of the corresponding rescaled processes. More precisely, for $\alpha\in (0,2)$, we prove that the rescaled processes, governed by time-inhomogeneous SDEs driven by additive processes, converge with respect to suitably chosen Wasserstein distances to time-homogeneous SDEs driven by symmetric $\alpha$-stable processes. Notably, the driving noise in the limiting SDEs depends only on large jumps of the underlying additive processes. A phase transition occurs for all values of the index $\alpha$, and a diffusive phenomenon arises whenever $\alpha\ge2$. |
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