Lyapunov exponents of linear stochastic functional-differential equations. II. Examples and case studies

Abstract

We give several examples and examine case studies of linear stochastic functional differential equations. The examples fall into two broad classes: regular and singular, according to whether an underlying stochastic semi-flow exists or not. In the singular case, we obtain upper and lower bounds on the maximal exponential growth rate $\overline{\lambda}_1 (\sigma)$ of the trajectories expressed in terms of the noise variance ƒÐ . Roughly speaking we show that for small $\sigma$, $\overline{\lambda}_1 (\sigma)$ behaves like $-\sigma^2 /2$, while for large $\sigma$, it grows like $\log \sigma$. In the regular case, it is shown that a discrete Oseledec spectrum exists, and upper estimates on the top exponent $\lambda_1$ are provided. These estimates are sharp in the sense that they reduce to known estimates in the deterministic or nondelay cases.