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arXiv 2209.10364math.PR
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平均法中的若干强极限定理

Some strong limit theorems in averaging

Yuri Kifer

备注：
arXiv admin note: text overlap with arXiv:2105.01940, arXiv:2111.05390
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AI总结：

本文研究快慢运动模型中平均过程的偏差收敛性，通过重定义过程证明了偏差过程与极限高斯过程的矩收敛阶及Prokhorov距离估计，在乘积情形下得到几乎必然收敛估计和函数形式重对数律。

摘要：

本文研究离散时间与连续时间下的快慢运动模型：离散时间模型为$X^ε((n+1)ε)=X^ε(nε)+εB(X^ε(nε),ξ(n))$，$n=0,1,...,[T/ε]$；连续时间模型为$\frac {dX^ε(t)}{dt}=B(X^ε(t),ξ(t/ε))$，$t∈[0,T]$，其中$B$是光滑向量函数，$ξ$是充分快混合的平稳随机过程。早在1966年Khasminskii就已证明：若$\bar X$为平均运动，则$G^ε=ε^{-1/2}(X^ε-\bar X)$弱收敛于高斯过程$G$。本文证明，对每个$ε$，可在足够丰富的概率空间上重新定义过程$ξ$和$G$且不改变其分布，使得$E\sup_{0≤t≤T}|G^ε(t)-G(t)|^{2M} =O(ε^δ)$（$δ>0$），由此还可得到$G^ε$与$G$的分布之间的Prokhorov距离估计为$O(ε^{δ/3})$。在乘积情形$B(x,ξ)=Σ(x)ξ$下，本文得到了形如$\begin{equation*}\begin{aligned}&\text{sup}\\&0≤t≤T \end{aligned}\end{equation*}|G^ε(t)-G(t)|=O(ε^δ)$的几乎必然收敛估计，以及$G^ε$的函数形式重对数律。本文指出，所采用的混合假设适用于由重要几类动力系统生成的快运动。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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