Averaging and mixing for stochastic perturbations of linear conservative systems
- Authors: Huang G.1,2, Kuksin S.B.3,2
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Affiliations:
- School of Physics, Beijing Institute of Technology
- Peoples Friendship University of Russia
- Université Paris VII — Denis Diderot, UFR de Mathématiques
- Issue: Vol 78, No 4 (2023)
- Pages: 3-52
- Section: Articles
- URL: https://journals.rcsi.science/0042-1316/article/view/133743
- DOI: https://doi.org/10.4213/rm10081
- ID: 133743
Cite item
Abstract
We study stochastic perturbations of linear systems of the form\begin{equation}dv(t)+Av(t) dt =\varepsilon P(v(t)) dt+\sqrt{\varepsilon} \mathcal{B}(v(t)) dW (t), \qquad v\in\mathbb{R}^D, \tag{*}\end{equation}where $A$ is a linear operator with non-zero imaginary spectrum.It is assumed that the vector field $P(v)$and the matrix function $\mathcal{B}(v)$ are locally Lipschitz with at most polynomial growth at infinity, that the equationis well-posed and a few of first moments of the norms of solutions $v(t)$ are bounded uniformly in $\varepsilon$. We use Khasminski'sapproach to stochastic averaging to show that, as $\varepsilon\to0$, a solution $v(t)$, written in the interaction representation interms of the operator $A$, for $0\leqslant t\leqslantConst\cdot\varepsilon^{-1}$ converges in distribution to a solution of an effective equation.The latter is obtained from $(*)$ by means of certain averaging. Assuming that equation $(*)$ and/or the effectiveequation are mixing, we examine this convergence further.Bibliography: 27 titles.
About the authors
Guan Huang
School of Physics, Beijing Institute of Technology; Peoples Friendship University of Russia
Email: huangguan@tsinghua.edu.cn
Sergei Borisovich Kuksin
Université Paris VII — Denis Diderot, UFR de Mathématiques; Peoples Friendship University of Russia
Author for correspondence.
Email: huangguan@tsinghua.edu.cn
Doctor of physico-mathematical sciences, Professor
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