论文标题
平均定理的一些很强的限制
Some strong limit theorems in averaging
论文作者
论文摘要
本文在离散时间$ x^ε（（（n+1）ε）= x^ε（nε）+εb（x^ε（nε），ξ（nε），ξ（n））$，$ n = 0,1，...，...，... {dx^ε（t）} {dt} = b（x^ε（t），ξ（t/ε））。\ \，t \ in [0，t] $，其中$ b $是平滑的向量函数，$ $ξ$是一个足够快的快速混合固定的随机过程。自1966年以来，人们知道，如果$ \ bar x $是平均运动，则$ g^ε=ε^{ - 1/2}（x^ε-\ bar x）$弱收敛到高斯流程$ g $。我们将证明，对于每个$ε$，过程$ξ$和$ g $都可以在足够丰富的概率空间上重新定义而不更改其发行量，以便$ e \ e \ sup_ {0 \ leq t \ leq t \ leq t} | g^ε（t）-g（t）-g（t）-g（t） $ o（ε^{δ/3}）$ prokhorov距离$ g^ε$和$ g $之间的距离。在产品案例中，$ b（x，ξ）=σ（x）ξ$我们获得了$ \ sup_ {0 \ leq t \ leq t} | g^ε（t）-g（t）-g（t）| = o（ε^δ）$ a.s。的$ \ sup_ {0 \ leq t \ leq t} | 我们注意到，我们的混合假设适用于重要的动态系统产生的快速运动。
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.
