Averaging principles for time-inhomogeneous multi-scale SDEs with partially dissipative coefficients

In this paper, we study averaging principles for a class of time-inhomogeneous stochastic differential equations (SDEs) with slow and fast time-scales, where the drift term in the fast component is time-dependent and only partially dissipative. Under asymptotic assumptions on the coefficients, we prove that the slow component $(X^{\varepsilon}_t)_{t\geq 0}$ converges strongly to the unique solution $(\bar{X}_t)_{t\geq 0}$ to an averaged SDE, when the diffusion coefficient in the slow component is independent of the fast component; on the other hand, we establish the weak convergence of $(X_t^{\varepsilon})_{t\ge0}$ in the space $C([0,T];\mathbb{R}^n)$ and identify the limiting process by the martingale problem approach, when the diffusion coefficient of the slow component depends on the fast component. The proofs of strong and weak averaging principles are partly based on the study of the existence and uniqueness of an evolution system of measures for time-inhomogeneous SDEs with partially dissipative drift.