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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2209.10364 |
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Table of Contents:
- The paper deals with the fast-slow motions setups in the discrete time $X^ε((n+1)ε)=X^ε(nε)+εB(X^ε(nε),ξ(n))$, $n=0,1,...,[T/ε]$ and the continuous time $\frac {dX^ε(t)}{dt}=B(X^ε(t),ξ(t/ε)).\, t\in [0,T]$ where $B$ is a smooth vector function and $ξ$ is a sufficiently fast mixing stationary stochastic process. It is known since 1966 (Khasminskii) that if $\bar X$ is the averaged motion then $G^ε=ε^{-1/2}(X^ε-\bar X)$ weakly converges to a Gaussian process $G$. We will show that for each $ε$ the processes $ξ$ and $G$ can be redefined on a sufficiently rich probability space without changing their distributions so that $E\sup_{0\leq t\leq T}|G^ε(t)-G(t)|^{2M} =O(ε^δ)$, $δ>0$ which gives also $O(ε^{δ/3})$ Prokhorov distance estimate between the distributions of $G^ε$ and $G$. In the product case $B(x,ξ)=Σ(x)ξ$ we obtain almost sure convergence estimates of the form $\sup_{0\leq t\leq T}|G^ε(t)-G(t)|=O(ε^δ)$ a.s., as well as the functional form of the law of iterated logarithm for $G^ε$. We note that our mixing assumptions are adapted to fast motions generated by important classes of dynamical systems.