Strong diffusion approximation in averaging

It is known since the 1960s (R. Khasminskii) that the slow motion $X^\varepsilon$ in the time-scaled multidimensional averaging setup $\frac {dX^\varepsilon(t)}{dt}=\frac 1\varepsilon B(X^\varepsilon(t), \xi(t/\varepsilon^2))+b(X^\varepsilon(t), \xi(t/\varepsilon^2)), t\in [0,T]$ converges weakly as $\varepsilon\to 0$ to a diffusion process provided $EB(x,\xi(s))\equiv 0$ where $\xi$ is a sufficiently fast mixing stochastic process when mixing is considered with respect to the $\sigma$-algebras generated by the process itself. The latter reduces substantially applications to dynamical systems (where $\xi(t)=T^t\omega$ for a flow $T^t$) and more recently I. Melbourne (Warwick) with various co-authors studied weak convergence under more applicable to dynamical systems assumptions which required use of the rough paths theory. I will discuss new results (some of them with P. Friz) about strong convergence in the above setups and their discrete time counterparts which yield new applications and some of them also rely on the rough paths theory. As a byproduct of this study we obtain almost sure invariance principles and then laws of iterated logarithm for iterated sums and iterated integrals.