Entrance measures for semigroups of time-inhomogeneous SDEs: possibly degenerate and expanding

In this article, we solve the problem of the long time behaviour of transition probabilities of time-inhomogeneous Markov processes and give a unified approach to stochastic differential equations (SDEs) with periodic, quasi-periodic, almost-periodic forcing and much beyond. We extend Harris's ``small set'' method to the time-inhomogeneous situation with the help of Hairer-Mattingly's refinement of Harris's recurrence to a one-step contraction of probability measures under the total variation distance $\rho_{\beta}$ weighted by some appropriate Lyapunov function and a constant $\beta>0$. We show that the convergence to an entrance measure under $\rho_{\beta}$ implies the convergence both in the total variation distance and the Wasserstein distance $\mathcal{W}_1$. For SDEs with locally Lipschitz and polynomial growth coefficients, in order to establish the local Doeblin condition, we obtain a nontrivial lower bound estimates for the fundamental solution of the corresponding Fokker-Planck equation. The drift term is allowed to be possibly non-weakly-dissipative, and the diffusion term can be degenerate over infinitely many large time intervals. This causes the system to be expanding over many periods of large time durations, the convergence to the entrance measure is generally only subgeometric. As an application we obtain the existence and uniqueness of quasi-periodic measure. We then lift the quasi-periodic Markovian semigroup to a cylinder on a torus and obtain a unique invariant measure and its ergodicity.