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We study Poisson equations and averaging for Stochastic DifferentialEquations (SDEs). Poisson equations are essential tools in both probabilitytheory and partial differential equations (PDEs). Their vast range ofapplications includes the study of the asymptotic behaviour of solutions ofparabolic PDEs, the treatment of multi-scale and homogenization problems aswell as the theoretical analysis of approximations of solution of StochasticDifferential Equations (SDEs). The study of Poisson equations in non-compactstate space is notoriously difficult. Results exists, but only for the casewhen the coefficients of the PDE are either bounded or satisfy linear growthassumptions (and even the latter case has only recently been achieved). In thispaper we treat Poisson equations on non-compact state spaces for coefficientsthat can grow super-linearly. This is one of the two building blocks towardsthe second (and main) result of the paper, namely in this paper we succeed inobtaining a {\em uniform in time} (UiT) averaging result (with a rate) for SDEmodels with super-linearly growing coefficients. This seems to be the first UiTaveraging result for slow-fast systems of SDEs (and indeed we are not aware ofany other UiT multiscale theorems in the PDE or ODE literature either). Key toobtaining both our UiT averaging result and to enable dealing with thesuper-linear growth of the coefficients (both of the slow-fast system and ofthe associated Poisson equation) is conquering exponential decay in time of thespace-derivatives of appropriate Markov semigroups. We refer to semigroupswhich enjoy this property as being Strongly Exponentially Stable.

Poisson Equations with locally-Lipschitz coefficients and Uniform in  Time Averaging for Stochastic Differential Equations via Strong Exponential  Stability