  A probabilistic approach for estimating sample qualities for stochastic
differential equations is introduced in this paper. The aim is to provide a
quantitative upper bound of the distance between the invariant probability
measure of a stochastic differential equation and that of its numerical
approximation. In order to extend estimates of finite time truncation error to
infinite time, it is crucial to know the rate of contraction of the transition
kernel of the SDE. We find that suitable numerical coupling methods can
effectively estimate such rate of contraction, which gives the distance between
two invariant probability measures. Our algorithms are tested with several low
and high dimensional numerical examples.
