Lyapunov exponents of nilpotent Itô systems with random coefficients. (Q1766002)

The stochastic differential equation \[ \begin{aligned} \text{d}v_ {t} = \begin{pmatrix} 0 & a(x_ {t}) \\ 0 & 0 \end{pmatrix} v_ {t}\,\text{d}t &+ \varepsilon ^ 2 \begin{pmatrix} a_ {11}(x_ {t}) & a_ {12}(x_ {t}) \\ a_ {21}(x_ {t}) & a_ {22}(x_ {t}) \end{pmatrix} v_ {t}\,\text{d}t \\ &+ \varepsilon \sum ^ {r}_ {\alpha =1} \begin{pmatrix} b^ \alpha _ {11}(x_ {t}) & b^ \alpha _ {12}(x_ {t}) \\ b^ \alpha _ {21}(x_ {t}) & b^ \alpha _ {22}(x_ {t}) \end{pmatrix} v_ {t}\,\text{d}W^ \alpha _ {t} \tag{1} \end{aligned} \] is considered, where \((W^ 1,\dots ,W^ {r})\) is a standard \(r\)-dimensional Wiener process and \(x\) is a diffusion process on a finite-dimensional \(\sigma \)-compact manifold \(M\), independent of the Wiener process. Suppose that the generator of the process \(x\) is elliptic, has smooth coefficients and there exists an invariant probability measure \(\mu \) for \(x\). Let the functions \(a\) and \(c(\cdot ) = \sum ^ {r}_ {\alpha =1} | b^ \alpha _ {21}| ^ 2\) be smooth and \(\mu \)-integrable, \(\bar a = \int _ {M} a\,\text{d}\mu \neq 0\) and \(\bar c = \int _ {M} c\,\text{d}\mu >0\). Under some additional technical hypotheses upon the functions \(a\), \(a_ {ij}\), \(b^ \alpha _ {ij}\) and the operator \(G\), it is shown that the top Lyapunov exponent \(\lambda _ \varepsilon \) of (1) satisfies \(\lambda _ \varepsilon = \varepsilon ^ {2/3}| \bar a| ^ {2/3} \bar c^ {1/3}\lambda _ 0 + O(\varepsilon ^ {4/3})\) as \(\varepsilon \to 0\), where \(\lambda _ 0\) is the top Lyapunov exponent of the equation \[ \text{d}v_ {t} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} v_ {t}\,\text{d}t + \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} v_ {t}\,\text{d}W_ {t}. \]