Strong diffusion approximation in averaging with dynamical systems fast motion (2105.01940v3)
Abstract: The paper deals with the fast-slow motions setups in the continuous time $\frac {dXt}{dt}=\frac 1\varepsilon B(X\varepsilon(t),\xi(t/\varepsilon2))+b(X\varepsilon(t),\,\xi(t/\varepsilon2)),\, t\in [0,T]$ and the discrete time $X\varepsilon((n+1)\varepsilon2)=X\varepsilon(n\varepsilon2)+\varepsilon B(X\varepsilon(n\varepsilon2),\xi(n)) +\varepsilon2 b(X\varepsilon(n\varepsilon2),\xi(n))$, $n=0,1,...,[T/\varepsilon2]$ 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 $\xi(t)=g(Ft\xi(0))$ or $\xi(n)=g(Fn\xi(0))$ 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 $\varepsilon$.