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The paper deals with the fast-slow motions setups in the continuous time$\frac {dX^\ve(t)}{dt}=\frac 1\ve\sig(X^\ve(t))\xi(t/\ve^2)+b(X^\ve(t)),\, t\in[0,T]$ and the discrete time$X_N((n+1)/N)=X_N(n/N)+N^{-1/2}\sig(X_N(n/N))\xi(n))+N^{-1}b(X_N(n/N))\xi(n)$,$n=0,1,...,[TN]$ where $\sig$ and $b$ are smooth matrix and vector functions,respectively, $\xi$ is a centered stationary vector stochastic process and$\ve, 1/N$ are small parameters. We derive, first, estimates in the stronginvariance principles for sums $S_{N}(t)=N^{-1/2}\sum_{0\leq k< [Nt]}\xi(k)$and iterated sums $\bbS^{ij}_{N}(t)=N^{-1}\sum_{0\leqk<l<[Nt]}\xi_i(k)\xi_j(l)$ together with the corresponding results forintegrals in the continuous time case which, in fact, yields almost sureinvariance principles for iterated sums and integrals of any order and,moreover, implies laws of iterated logarithm for these objects. Then, relyingon the rough paths theory, we obtain strong almost sure approximations ofprocesses $X^\ve$ and $X_N$ by corresponding diffusion processes $\Xi^\ve$ and$\Xi_N$, respectively. Previous results for the above setup dealt either withweak or moment diffusion approximations and not with almost sure approximationwhich is the new and natural generalization of well known works on stronginvariance principles for sums of weakly dependent random variables.

Almost sure diffusion approximation in averaging via rough paths theory