Strong diffusion approximation in averaging with dynamical systems fast motion

Abstract

The paper deals with the fast-slow motions setups in the continuous time $\frac {dX^(t)}{dt}=\frac 1\varepsilon B(X^\varepsilon(t),\xi(t/\varepsilon^2))+b(X^\varepsilon(t),\,\xi(t/\varepsilon^2)),\, t\in [0,T]$ and the discrete time $X^\varepsilon((n+1)\varepsilon^2)=X^\varepsilon(n\varepsilon^2)+\varepsilon B(X^\varepsilon(n\varepsilon^2),\xi(n)) +\varepsilon^2 b(X^\varepsilon(n\varepsilon^2),\xi(n))$, $n=0,1,...,[T/\varepsilon^2]$ where $\Sigma$ and $b$ are smooth vector functions and $\xi$ 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 $X^\varepsilon$ and a family of diffusions $\Xi^\varepsilon$ can be redefined on a common sufficiently rich probability space so that $E\sup_{0\leq t\leq T}|X^\varepsilon(t)-\Xi^\varepsilon(t)|^{p}\leq C\varepsilon^\delta,\, p\geq 1$ for some $C,\delta>0$ and all $\varepsilon>0$, where all $\Xi^\varepsilon,\, \varepsilon>0$ have the same diffusion coefficients but underlying Brownian motions may change with $\varepsilon$.