  This paper proposes a new multilevel Monte Carlo (MLMC) method for the
ergodic SDEs which do not satisfy the contractivity condition. By introducing
the change of measure technique, we simulate the path with contractivity and
add the Radon-Nykodim derivative to the estimator. We can show the strong error
of the path is uniformly bounded with respect to $T.$ Moreover, the variance of
the new level estimators increase linearly in $T,$ which is a great reduction
compared with the exponential increase in standard MLMC. Then the total
computational cost is reduced to $O(\varepsilon^{-2}|\log \varepsilon|^{2})$
from $O(\varepsilon^{-3}|\log \varepsilon|)$ of the standard Monte Carlo
method. Numerical experiments support our analysis.
