Published in Annals of Probability, 25(3), 1210-1240.


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 $\overlineλ1$(σ) of the trajectories expressed in terms of the noise variance σ . Roughly speaking we show that for small σ, $\overlineλ1$(σ) behaves like -σ2 /2, while for large σ, it grows like logσ. In the regular case, it is shown that a discrete Oseledec spectrum exists, and upper estimates on the top exponent λ1 are provided. These estimates are sharp in the sense that they reduce to known estimates in the deterministic or nondelay cases.