Strong diffusion approximation in averaging with dynamical systems fast motion

TL;DR

This paper establishes strong diffusion approximations for fast-slow dynamical systems under broad conditions, showing that the slow component can be closely approximated by a diffusion process with explicit error bounds.

Contribution

It extends diffusion approximation results to a wider class of dynamical systems with hyperbolic features, allowing for applications to more general observables and stochastic processes.

Findings

Strong approximation of $X^ ext{epsilon}$ by diffusion processes with explicit error bounds

Applicable to systems with hyperbolic dynamics and broad classes of observables

Diffusions share the same coefficients but may have different Brownian motions for each epsilon

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 $\varepsilon$.