The Lyapunov exponent for a general Sturm-Liouville operator with random coefficients. I: Small disorder (Q2640230)

Given the Sturm-Liouville type random operator \[ Hy=-(1+\epsilon f_ 1(x_ t))^{-1}(d/dt)((1+\epsilon f_ 2(x_ t))^{- 1}(dy/dt))+\epsilon_ 1F(x_ t), \] where \(x_ t\) is Brownian motion on a compact Riemannian manifold M, F, \(f_ 1\) and \(f_ 2\) are smooth nonconstant functions from M to \({\mathbb{R}}\), and \(\epsilon,\epsilon_ 1>0\). The first term of the asymptotic expansion of the top Lyapunov exponent and the rotation number of \(Hy=\lambda y\), \(\lambda >0\), is obtained for \(\epsilon =\epsilon_ 1\to 0\), \(\epsilon_ 1=o(\epsilon)\) and \(\epsilon\to 0\), and for \(\epsilon =o(\epsilon_ 1)\) and \(\epsilon_ 1\to 0\). The expressions are in terms of the expansions of F and \(f_ 2-f_ 1\) w.r.t. the eigenfunctions of \(\Delta\) \(=\) Laplace- Beltrami operator on M. The results of \textit{L. Arnold}, \textit{G. Papanicolaou} and \textit{V. Wihstutz} [SIAM J. Appl. Math. 46, 427-450 (1986; Zbl 0603.60051)] are recovered and generalized.