#### 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 $\overlineλ_{1}$(σ) of the trajectories expressed in terms of the noise variance σ . Roughly speaking we show that for small σ, $\overlineλ_{1}$(σ) behaves like -σ^{2} /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 λ_{1} are provided. These estimates are sharp in the sense that they reduce to known estimates in the deterministic or nondelay cases.

#### Recommended Citation

Mohammed, Salah-Eldin A. and Scheutzow, Michael K. "Lyapunov Exponents of Linear Stochastic Functional-Differential Equations. II. Examples and Case Studies." (Jan 1997).

#### Included in

Mathematics Commons, Ordinary Differential Equations and Applied Dynamics Commons, Probability Commons

## Comments

Published in

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