  We consider the long time behavior of Wong-Zakai approximations of stochastic
differential equations. These piecewise smooth diffusion approximations are of
great importance in many areas, such as those with ordinary differential
equations associated to random smooth fluctuations; e.g. robust filtering
problems. In many examples, the mean error estimate bounds that have been
derived in the literature can grow exponentially with respect to the time
horizon. We show in a simple example that indeed mean error estimates do
explode exponentially in the time parameter, i.e. in that case a Wong-Zakai
approximation is only useful for extremely short time intervals. Under spectral
conditions, we present some quantitative time-uniform convergence theorems,
i.e. time-uniform mean error bounds, yielding what seems to be the first
results of this type for Wong-Zakai diffusion approximations.
