  A new explicit stabilized scheme of weak order one for stiff and ergodic
stochastic differential equations (SDEs) is introduced. In the absence of
noise, the new method coincides with the classical deterministic stabilized
scheme (or Chebyshev method) for diffusion dominated advection-diffusion
problems and it inherits its optimal stability domain size that grows
quadratically with the number of internal stages of the method. For mean-square
stable stiff stochastic problems, the scheme has an optimal extended
mean-square stability domain that grows at the same quadratic rate as the
deterministic stability domain size in contrast to known existing methods for
stiff SDEs [A. Abdulle and T. Li. Commun. Math. Sci., 6(4), 2008, A. Abdulle,
G. Vilmart, and K. C. Zygalakis, SIAM J. Sci. Comput., 35(4), 2013]. Combined
with postprocessing techniques, the new methods achieve a convergence rate of
order two for sampling the invariant measure of a class of ergodic SDEs,
achieving a stabilized version of the non-Markovian scheme introduced in [B.
Leimkuhler, C. Matthews, and M. V. Tretyakov, Proc. R. Soc. A, 470, 2014].
