Some strong limit theorems in averaging

arXiv2209.10364MaRDI QIDQ6411484

Yuri Kifer

Publication date: 21 September 2022

Abstract: The paper deals with the fast-slow motions setups in the discrete time $X^{e}psilon((n+1)epsilon)=X^{e}psilon(nepsilon)+epsilonB(X^{e}psilon(nepsilon),xi(n))$, $n=0,1,...,[T/epsilon]$ and the continuous time $fracd{X}^{e}psilon(t)dt=B(X^{e}psilon(t),xi(t/epsilon)).,tin[0,T]$ where $B$ is a smooth vector function and $xi$ is a sufficiently fast mixing stationary stochastic process. It is known since 1966 (Khasminskii) that if $$ is the averaged motion then $$ weakly converges to a Gaussian process $G$. We will show that for each $epsilon$ the processes $xi$ and $G$ can be redefined on a sufficiently rich probability space without changing their distributions so that $Esu{p}_{0leqtleqT}|G^{e}psilon(t)-G(t){|}^{2M}=O(epsilo{n}^{delta})$, $delta>0$ which gives also $O(epsilo{n}^{delta/3})$ Prokhorov distance estimate between the distributions of $G^{e}psilon$ and $G$. In the product case $B(x,xi)=Sigma(x)xi$ we obtain almost sure convergence estimates of the form $su{p}_{0leqtleqT}|G^{e}psilon(t)-G(t)|=O(epsilo{n}^{d}elta)$ a.s., as well as the functional form of the law of iterated logarithm for $G^{e}psilon$. We note that our mixing assumptions are adapted to fast motions generated by important classes of dynamical systems.

Mathematics Subject Classification ID

Strong limit theorems (60F15) Averaging method for ordinary differential equations (34C29)

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