Strong diffusion approximation in averaging with dynamical systems fast motion

The paper deals with the fast-slow motions setups in the continuous time $\frac {dX^(t)}{dt}=\frac 1\varepsilon B(X^\varepsilon(t),\xi(t/\varepsilon^2))+b(X^\varepsilon(t),,\xi(t/\varepsilon^2)),, t\in [0,T]$andthediscretetime$X^\varepsilon((n+1)\varepsilon^2)=X^\varepsilon(n\varepsilon^2)+\varepsilon B(X^\varepsilon(n\varepsilon^2),\xi(n)) +\varepsilon^2 b(X^\varepsilon(n\varepsilon^2),\xi(n))$,$n=0,1,…,[T/\varepsilon^2]$where$\Sigma$and$b$aresmoothvectorfunctionsand$\xi$ is a stationary vector stochastic process such that $E\xi (0)=0$ for all $x\in {\mathbb{R}}^d$. Unlike \cite{Ki20} the assumptions imposed on the process $\xi$ allow applications to a wide class of observables $g$ in the dynamical systems setup so that $\xi$ can be taken in the form $\xi (t)=g(F^t\xi (0))$ or $\xi (n)=g(F^n\xi (0))$ where $F$ is either a flow or a diffeomorphism with some hyperbolicity and $g$ is a vector function. In this paper we show that both $X^{\epsilon }$ and a family of diffusions ${\mathrm{\Xi }}^{\epsilon }$ can be redefined on a common sufficiently rich probability space so that $E\sup_{0\leq t\leq T}|X^\varepsilon(t)-\Xi^\varepsilon(t)|^{p}\leq C\varepsilon^\delta,, p\geq 1$forsome$C,\delta>0$andall$\varepsilon>0$,whereall$\Xi^\varepsilon,, \varepsilon>0$ have the same diffusion coefficients but underlying Brownian motions may change with $\epsilon$.