Almost sure diffusion approximation in averaging via rough paths theory

Abstract: 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 strong invariance principles for sums $S_{N}(t)=N^{-1/2}\sum_{0\leq k< [Nt]}\xi(k)$ and iterated sums $\bbS^{ij}{N}(t)=N^{-1}\sum{0\leq k<l<[Nt]}\xi_i(k)\xi_j(l)$ together with the corresponding results for integrals in the continuous time case which, in fact, yields almost sure invariance principles for iterated sums and integrals of any order and, moreover, implies laws of iterated logarithm for these objects. Then, relying on the rough paths theory, we obtain strong almost sure approximations of processes $X^\ve$ and $X_N$ by corresponding diffusion processes $\Xi^\ve$ and $\Xi_N$, respectively. Previous results for the above setup dealt either with weak or moment diffusion approximations and not with almost sure approximation which is the new and natural generalization of well known works on strong invariance principles for sums of weakly dependent random variables.