Strong averaging principle for multiscale time-inhomogeneous SDEs with multiplicative $\alpha$-stable noises

Abstract

In this paper, we study the strong averaging principle for multiscale time-inhomogeneous stochastic systems driven by multiplicative $\alpha$-stable processes with $\alpha\in(1,2)$. Based on Khasminskii's discretization approach, we first establish that the fast component processes with a frozen slow variable admits a periodic measure. We then prove the strong convergence of the slow subsystem to an averaged system that depends on the time scale $\varepsilon$. For any fixed $\varepsilon$, if the reciprocals of the two periods $\tau_1$ and $\varepsilon \tau_2$ are rationally linearly independent, an important consequence is that the averaged system has random quasi-periodicity. Furthermore, by applying the ergodic theorem, we prove the strong convergence of the slow subsystem to another averaged system, a time-inhomogeneous SDEs independent of the time scale $\varepsilon$. Our result is also novel even in the time-homogeneous case for a fully coupled multiscale system with multiplicative $\alpha$-stable noises. Finally, we apply the result to a climate-weather system.