  We study averaging for Stochastic Differential Equations (SDEs) and Poisson
equations. We succeed in obtaining a uniform in time (UiT) averaging result,
with a rate, for fully coupled SDE models with super-linearly growing
coefficients. This is the main result of this paper and is, to the best of our
knowledge, the first UiT multiscale result with a rate. Very few UiT averaging
results exist in the literature, and they almost exclusively apply to
multiscale systems of Ordinary Differential Equations. Among these few, none of
those we are aware of comes with a rate of convergence. The UiT nature of this
result and the rate of convergence given by the main theorem, make it important
as theoretical underpinning for a range of applications, such as applications
to statistical methodology, molecular dynamics etc. Key to obtaining both our
UiT averaging result and to enable dealing with the super-linear growth of the
coefficients is conquering exponential decay in time of the space-derivatives
of appropriate Markov semigroups. We refer to this property as being Strongly
Exponentially Stable (SES). The analytic approach to proving averaging results
we take requires studying a family of Poisson problems associated with the
generator of the (fast component of the) SDE dynamics. The study of Poisson
equations in non-compact state space is notoriously difficult, with current
literature mostly covering the case when the coefficients of the Partial
Differential Equation (PDE) are either bounded or satisfy linear growth
assumptions. In this paper we treat Poisson equations on non-compact state
spaces for coefficients that can grow super-linearly. We demonstrate how SES
can be employed not only to prove the UiT result for the slow-fast system but
also to overcome some of the technical hurdles in the analysis of Poisson
problems, which is of independent interest as well.
