  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.
