Strong diffusion approximation in averaging with dynamical systems fast motion

The paper deals with the fast-slow motions setups in the continuous time $\frac {dX^(t)}{dt}=\frac 1\varepsilon B(X^\varepsilon(t),ξ(t/\varepsilon^2))+b(X^\varepsilon(t),\,ξ(t/\varepsilon^2)),\, t\in [0,T]$ and the discrete time $X^\varepsilon((n+1)\varepsilon^2)=X^\varepsilon(n\varepsilon^2)+\varepsilon B(X^\varepsilon(n\varepsilon^2),ξ(n)) +\varepsilon^2 b(X^\varepsilon(n\varepsilon^2),ξ(n))$, $n=0,1,...,[T/\varepsilon^2]$ where $Σ$ and $b$ are smooth vector functions and $ξ$ is a stationary vector stochastic process such that $Eξ(0)=0$ for all $x\in\mathbb{R}^d$. Unlike \cite{Ki20} the assumptions imposed on the process $ξ$ allow applications to a wide class of observables $g$ in the dynamical systems setup so that $ξ$ can be taken in the form $ξ(t)=g(F^tξ(0))$ or $ξ(n)=g(F^nξ(0))$ 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 $X^\varepsilon$ and a family of diffusions $Ξ^\varepsilon$ can be redefined on a common sufficiently rich probability space so that $E\sup_{0\leq t\leq T}|X^\varepsilon(t)-Ξ^\varepsilon(t)|^{p}\leq C\varepsilon^δ,\, p\geq 1$ for some $C,δ>0$ and all $\varepsilon>0$, where all $Ξ^\varepsilon,\, \varepsilon>0$ have the same diffusion coefficients but underlying Brownian motions may change with $\varepsilon$.

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