论文标题
平均和混合线性保守系统的随机扰动
Averaging and mixing for stochastic perturbations of linear conservative systems
论文作者
论文摘要
我们研究了$$ dv（t）+av（t）dt =εp（v（t））dt+\sqrtεb（v（t））dw（t），v \ in \ mathbb {r}^{d}^{d}，（*）$ a $ a linear-imection a in-inmecriance clothimen的随机扰动。假定矢量场$ p（v）$和矩阵函数$ b（v）$是本地Lipschitz，最多最多是无穷大的多项式增长，方程式是很好的，并且在$ε$中均匀地构成了解决方案规范$ v（t）$的第一瞬间。我们使用Khasminski方法进行随机平均值，以表明AS $ε\ to0 $，解决方案$ v（t）$，以互动表示用操作员$ a $表示，以$ 0 \ le t \ le t \ le t \ le t \ le const \，ε^{ - 1} $收敛到分布有效方程的解决方案。后者是通过某些平均值从（*）获得的。假设等式（*）和/或有效方程式正在混合，我们会进一步研究这种收敛。
We study stochastic perturbations of linear systems of the form $$ dv(t)+Av(t)dt = εP(v(t))dt+\sqrtεB(v(t)) dW (t), v\in\mathbb{R}^{D}, (*) $$ where $A$ is a linear operator with non-zero imaginary spectrum. It is assumed that the vector field $P(v)$ and the matrix-function $B(v)$ are locally Lipschitz with at most a polynomial growth at infinity, that the equation is well posed and first few moments of norms of solutions $v(t)$ are bounded uniformly in $ε$. We use the Khasminski approach to stochastic averaging to show that as $ε\to0$, a solution $v(t)$, written in the interaction representation in terms of operator $A$, for $0\le t \le Const\,ε^{-1}$ converges in distribution to a solution of an effective equation. The latter is obtained from (*) by means of certain averaging. Assuming that eq.(*) and/or the effective equation are mixing, we examine this convergence further.
